4.2 Ordinary & Partial Differentiation

Detailed Theory: Ordinary & Partial Differentiation

1. Introduction to Derivatives

1.1 What is a Derivative?

The derivative measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line to the curve at a point.

1.2 Definition of Derivative

The derivative of a function $f(x)$ at point $x = a$ is:

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Alternative notation: $f'(x)$, $\frac{df}{dx}$, $\frac{d}{dx}f(x)$

1.3 Geometric Interpretation

• $f'(a)$ = slope of tangent line to $y = f(x)$ at $x = a$

• If $f'(a) > 0$: function increasing at $x = a$

• If $f'(a) < 0$: function decreasing at $x = a$

• If $f'(a) = 0$: possible local maximum/minimum or inflection point

1.4 Example: Finding Derivative from Definition

Find derivative of $f(x) = x^2$ using definition.

Solution:

$$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$$

$$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$$

$$= \lim_{h \to 0} \frac{2xh + h^2}{h}$$

$$= \lim_{h \to 0} (2x + h) = 2x$$

So $f'(x) = 2x$

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2. Basic Differentiation Rules

2.1 Power Rule

For any real number $n$:

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Examples:

• $\frac{d}{dx}(x^3) = 3x^2$

• $\frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

• $\frac{d}{dx}\left(\frac{1}{x}\right) = \frac{d}{dx}(x^{-1}) = -x^{-2} = -\frac{1}{x^2}$

2.2 Constant Rule

For constant $c$:

$$\frac{d}{dx}(c) = 0$$

2.3 Constant Multiple Rule

For constant $c$:

$$\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$$

Example: $\frac{d}{dx}(5x^3) = 5 \cdot 3x^2 = 15x^2$

2.4 Sum/Difference Rule

$$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$$

Example: $\frac{d}{dx}(x^2 + 3x - 5) = 2x + 3$

2.5 Product Rule

$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$$

Example: Find $\frac{d}{dx}(x^2 \sin x)$

$$\frac{d}{dx}(x^2 \sin x) = (2x)(\sin x) + (x^2)(\cos x) = 2x \sin x + x^2 \cos x$$

2.6 Quotient Rule

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \quad (g(x) \neq 0)$$

Example: Find $\frac{d}{dx}\left(\frac{x}{x^2 + 1}\right)$

$$\frac{d}{dx}\left(\frac{x}{x^2 + 1}\right) = \frac{(1)(x^2 + 1) - (x)(2x)}{(x^2 + 1)^2}$$

$$= \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}$$

2.7 Chain Rule

For composite function $f(g(x))$:

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Alternative notation: If $y = f(u)$ and $u = g(x)$, then:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Example: Find $\frac{d}{dx}[\sin(x^2)]$

Let $u = x^2$, then $y = \sin u$

$$\frac{dy}{du} = \cos u = \cos(x^2)$$

$$\frac{du}{dx} = 2x$$

So $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x \cos(x^2)$

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3. Derivatives of Elementary Functions

3.1 Trigonometric Functions

1. $\frac{d}{dx}(\sin x) = \cos x$

2. $\frac{d}{dx}(\cos x) = -\sin x$

3. $\frac{d}{dx}(\tan x) = \sec^2 x$

4. $\frac{d}{dx}(\cot x) = -\csc^2 x$

5. $\frac{d}{dx}(\sec x) = \sec x \tan x$

6. $\frac{d}{dx}(\csc x) = -\csc x \cot x$

3.2 Exponential and Logarithmic Functions

1. $\frac{d}{dx}(e^x) = e^x$

2. $\frac{d}{dx}(a^x) = a^x \ln a \quad (a > 0, a \neq 1)$

3. $\frac{d}{dx}(\ln x) = \frac{1}{x} \quad (x > 0)$

4. $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} \quad (a > 0, a \neq 1)$

3.3 Inverse Trigonometric Functions

1. $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} \quad (-1 < x < 1)$

2. $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \quad (-1 < x < 1)$

3. $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}$

4. $\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2}$

5. $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2 - 1}} \quad (|x| > 1)$

6. $\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2 - 1}} \quad (|x| > 1)$

3.4 Hyperbolic Functions

1. $\frac{d}{dx}(\sinh x) = \cosh x$

2. $\frac{d}{dx}(\cosh x) = \sinh x$

3. $\frac{d}{dx}(\tanh x) = \text{sech}^2 x$

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4. Higher Order Derivatives

4.1 Definition

The second derivative is the derivative of the first derivative:

$$f''(x) = \frac{d}{dx}[f'(x)] = \frac{d^2f}{dx^2}$$

The n-th derivative:

$$f^{(n)}(x) = \frac{d^n f}{dx^n}$$

4.2 Notation

• First derivative: $f'(x)$, $f^{(1)}(x)$, $\frac{df}{dx}$

• Second derivative: $f''(x)$, $f^{(2)}(x)$, $\frac{d^2f}{dx^2}$

• n-th derivative: $f^{(n)}(x)$, $\frac{d^n f}{dx^n}$

4.3 Physical Interpretation

• Position → Velocity → Acceleration:

  • If $s(t)$ = position, then $v(t) = s'(t)$ = velocity

  • $a(t) = v'(t) = s''(t)$ = acceleration

• Curvature: Second derivative gives information about concavity

  • $f''(x) > 0$: concave up

  • $f''(x) < 0$: concave down

  • $f''(x) = 0$: possible inflection point

4.4 Example

Find first three derivatives of $f(x) = x^3 - 3x^2 + 2x$

$$f'(x) = 3x^2 - 6x + 2$$

$$f''(x) = 6x - 6$$

$$f'''(x) = 6$$

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5. Implicit Differentiation

5.1 What is Implicit Differentiation?

Used when $y$ is defined implicitly as a function of $x$ by an equation, not explicitly as $y = f(x)$.

5.2 Method

1. Differentiate both sides of equation with respect to $x$

2. Treat $y$ as function of $x$ (use chain rule for $y$ terms)

3. Solve for $\frac{dy}{dx}$

5.3 Examples

Example 1: Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$

Differentiate both sides:

$$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)$$

$$2x + 2y \frac{dy}{dx} = 0$$

Solve for $\frac{dy}{dx}$:

$$2y \frac{dy}{dx} = -2x$$

$$\frac{dy}{dx} = -\frac{x}{y}$$

Example 2: Find $\frac{dy}{dx}$ for $x^3 + y^3 = 6xy$

Differentiate:

$$3x^2 + 3y^2 \frac{dy}{dx} = 6y + 6x \frac{dy}{dx}$$

Rearrange:

$$3y^2 \frac{dy}{dx} - 6x \frac{dy}{dx} = 6y - 3x^2$$

$$(3y^2 - 6x) \frac{dy}{dx} = 6y - 3x^2$$

$$\frac{dy}{dx} = \frac{6y - 3x^2}{3y^2 - 6x} = \frac{2y - x^2}{y^2 - 2x}$$

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6. Logarithmic Differentiation

6.1 When to Use

Useful for:

1. Functions of form $y = [f(x)]^{g(x)}$

2. Products/quotients with many factors

3. Functions with variables in both base and exponent

6.2 Method

1. Take natural logarithm of both sides: $\ln y = \ln[f(x)]$

2. Simplify using logarithm properties

3. Differentiate implicitly with respect to $x$

4. Solve for $\frac{dy}{dx}$

6.3 Examples

Example 1: Find $\frac{dy}{dx}$ for $y = x^x$

Take natural log: $\ln y = \ln(x^x) = x \ln x$

Differentiate implicitly:

$$\frac{1}{y} \frac{dy}{dx} = \ln x + x \cdot \frac{1}{x} = \ln x + 1$$

$$\frac{dy}{dx} = y(\ln x + 1) = x^x(\ln x + 1)$$

Example 2: Find $\frac{dy}{dx}$ for $y = \frac{(x+1)^2 \sqrt{x-1}}{(x+3)^3}$

Take natural log:

$$\ln y = 2\ln(x+1) + \frac{1}{2}\ln(x-1) - 3\ln(x+3)$$

Differentiate:

$$\frac{1}{y} \frac{dy}{dx} = \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3}$$

$$\frac{dy}{dx} = y\left[\frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3}\right]$$

$$= \frac{(x+1)^2 \sqrt{x-1}}{(x+3)^3} \left[\frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3}\right]$$

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7. Applications of Derivatives

7.1 Tangent and Normal Lines

• Tangent line at $x = a$: $y - f(a) = f'(a)(x - a)$

• Normal line at $x = a$: $y - f(a) = -\frac{1}{f'(a)}(x - a)$ (if $f'(a) \neq 0$)

Example: Find tangent and normal to $y = x^2$ at $(2, 4)$

$f'(x) = 2x$, so $f'(2) = 4$

Tangent: $y - 4 = 4(x - 2)$ or $y = 4x - 4$

Normal: $y - 4 = -\frac{1}{4}(x - 2)$ or $y = -\frac{1}{4}x + \frac{9}{2}$

7.2 Rates of Change

If $y = f(x)$, then $\frac{dy}{dx}$ = instantaneous rate of change of $y$ with respect to $x$.

Example: Radius of circle increasing at 3 cm/s. How fast is area changing when radius = 10 cm?

$A = \pi r^2$, $\frac{dr}{dt} = 3$

$$\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} = 2\pi r \cdot 3 = 6\pi r$$

When $r = 10$: $\frac{dA}{dt} = 6\pi(10) = 60\pi \text{ cm}^2/\text{s}$

7.3 Related Rates

Problems where two or more quantities are related and changing over time.

Strategy:

1. Identify given rates and wanted rate

2. Find equation relating variables

3. Differentiate with respect to time

4. Substitute known values and solve

Example: Ladder 10 ft long slides down wall. Bottom moves away at 1 ft/s. How fast is top sliding down when bottom is 6 ft from wall?

Let $x$ = distance from wall, $y$ = height on wall

Given: $x^2 + y^2 = 100$, $\frac{dx}{dt} = 1$, find $\frac{dy}{dt}$ when $x = 6$

Differentiate: $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$

When $x = 6$, $y = \sqrt{100 - 36} = \sqrt{64} = 8$

$$2(6)(1) + 2(8)\frac{dy}{dt} = 0$$

$$12 + 16\frac{dy}{dt} = 0$$

$$\frac{dy}{dt} = -\frac{12}{16} = -\frac{3}{4} \text{ ft/s}$$

Negative means top is sliding down.

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8. Partial Differentiation

8.1 Functions of Several Variables

A function of two variables: $z = f(x, y)$

Example: $f(x, y) = x^2 + 2xy + y^2$

8.2 Partial Derivatives

• Partial derivative with respect to $x$: Treat $y$ as constant Notation: $f_x$, $\frac{\partial f}{\partial x}$, $\frac{\partial}{\partial x}f(x, y)$

• Partial derivative with respect to $y$: Treat $x$ as constant Notation: $f_y$, $\frac{\partial f}{\partial y}$, $\frac{\partial}{\partial y}f(x, y)$

8.3 Computing Partial Derivatives

Example: For $f(x, y) = x^2y + \sin(xy)$

$$f_x = \frac{\partial f}{\partial x} = 2xy + y\cos(xy)$$

$$f_y = \frac{\partial f}{\partial y} = x^2 + x\cos(xy)$$

8.4 Higher Order Partial Derivatives

For $f(x, y)$:

1. Second partials:

   • $f_{xx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^2 f}{\partial x^2}$

   • $f_{yy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial^2 f}{\partial y^2}$

   • $f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^2 f}{\partial y \partial x}$

   • $f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial^2 f}{\partial x \partial y}$

2. Clairaut's Theorem: If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$

8.5 Example: Higher Order Partials

For $f(x, y) = e^{x^2 + y}$

$$f_x = 2xe^{x^2 + y}$$

$$f_y = e^{x^2 + y}$$

$$f_{xx} = \frac{\partial}{\partial x}(2xe^{x^2 + y}) = 2e^{x^2 + y} + 4x^2e^{x^2 + y} = 2e^{x^2 + y}(1 + 2x^2)$$

$$f_{yy} = \frac{\partial}{\partial y}(e^{x^2 + y}) = e^{x^2 + y}$$

$$f_{xy} = \frac{\partial}{\partial y}(2xe^{x^2 + y}) = 2xe^{x^2 + y}$$

$$f_{yx} = \frac{\partial}{\partial x}(e^{x^2 + y}) = 2xe^{x^2 + y}$$

Note: $f_{xy} = f_{yx}$

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9. Total Differential and Chain Rule for Partial Derivatives

9.1 Total Differential

For $z = f(x, y)$, the total differential is:

$$dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy$$

Interpretation: Approximate change in $z$ when $x$ changes by $dx$ and $y$ changes by $dy$.

9.2 Chain Rule for Partial Derivatives

Case 1: $z = f(x, y)$, $x = g(t)$, $y = h(t)$

$$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$$

Case 2: $z = f(x, y)$, $x = g(s, t)$, $y = h(s, t)$

$$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s}$$

$$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t}$$

9.3 Example of Chain Rule

Let $z = x^2y$, $x = s + t$, $y = st$

Find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$

First: $\frac{\partial z}{\partial x} = 2xy$, $\frac{\partial z}{\partial y} = x^2$

Also: $\frac{\partial x}{\partial s} = 1$, $\frac{\partial x}{\partial t} = 1$, $\frac{\partial y}{\partial s} = t$, $\frac{\partial y}{\partial t} = s$

Now:

$$\frac{\partial z}{\partial s} = (2xy)(1) + (x^2)(t) = 2xy + tx^2$$

Substitute $x = s+t$, $y = st$:

$$= 2(s+t)(st) + t(s+t)^2 = 2s^2t + 2st^2 + t(s^2 + 2st + t^2)$$

$$= 2s^2t + 2st^2 + ts^2 + 2st^2 + t^3 = 3s^2t + 4st^2 + t^3$$

Similarly:

$$\frac{\partial z}{\partial t} = (2xy)(1) + (x^2)(s) = 2xy + sx^2$$

Substitute: $= 2(s+t)(st) + s(s+t)^2 = 2s^2t + 2st^2 + s^3 + 2s^2t + st^2$

$$= s^3 + 4s^2t + 3st^2$$

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10. Gradient and Directional Derivatives

10.1 Gradient Vector

For $f(x, y)$, the gradient is:

$$\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle$$

For $f(x, y, z)$:

$$\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$$

10.2 Properties of Gradient

1. Points in direction of maximum increase of $f$

2. Magnitude = maximum rate of increase

3. Perpendicular to level curves/surfaces

10.3 Directional Derivative

Rate of change of $f$ in direction of unit vector $\vec{u} = \langle a, b \rangle$:

$$D_{\vec{u}} f = \nabla f \cdot \vec{u} = f_x a + f_y b$$

Example: Find directional derivative of $f(x, y) = x^2 + y^2$ at $(1, 2)$ in direction of $\vec{v} = \langle 3, 4 \rangle$

First, make $\vec{v}$ unit vector: $\|\vec{v}\| = \sqrt{9 + 16} = 5$

$$\vec{u} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle$$

Gradient: $\nabla f = \langle 2x, 2y \rangle$

At $(1, 2)$: $\nabla f(1, 2) = \langle 2, 4 \rangle$

Directional derivative:

$$D_{\vec{u}} f = \langle 2, 4 \rangle \cdot \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle = \frac{6}{5} + \frac{16}{5} = \frac{22}{5}$$

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11. Taylor Series and Maclaurin Series

11.1 Taylor Series for Single Variable

For $f(x)$ about $x = a$:

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$

$$= \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

11.2 Maclaurin Series

Taylor series about $a = 0$:

$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$

11.3 Important Maclaurin Series

1. $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

2. $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

3. $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

4. $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} \quad (|x| < 1)$

11.4 Taylor Series for Two Variables

For $f(x, y)$ about $(a, b)$:

$$f(x, y) = f(a, b) + f_x(a, b)(x-a) + f_y(a, b)(y-b)$$

$$+ \frac{1}{2!}[f_{xx}(a, b)(x-a)^2 + 2f_{xy}(a, b)(x-a)(y-b) + f_{yy}(a, b)(y-b)^2] + \cdots$$

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12. Applications of Partial Derivatives

12.1 Tangent Plane to Surface

For surface $z = f(x, y)$ at point $(x_0, y_0, z_0)$:

Equation of tangent plane:

$$z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$$

Example: Find tangent plane to $z = x^2 + y^2$ at $(1, 2, 5)$

$$f_x = 2x \Rightarrow f_x(1, 2) = 2$$

$$f_y = 2y \Rightarrow f_y(1, 2) = 4$$

Tangent plane: $z - 5 = 2(x - 1) + 4(y - 2)$

$$z = 2x - 2 + 4y - 8 + 5 = 2x + 4y - 5$$

12.2 Linear Approximation

For $f(x, y)$ near $(a, b)$:

$$L(x, y) = f(a, b) + f_x(a, b)(x-a) + f_y(a, b)(y-b)$$

Approximation: $f(x, y) \approx L(x, y)$ for $(x, y)$ near $(a, b)$

12.3 Optimization

To find local maxima/minima of $f(x, y)$:

1. Find critical points: Solve $f_x = 0$, $f_y = 0$

2. Use Second Derivative Test: Let $D = f_{xx}f_{yy} - (f_{xy})^2$ at critical point $(a, b)$

   • If $D > 0$ and $f_{xx} > 0$: Local minimum

   • If $D > 0$ and $f_{xx} < 0$: Local maximum

   • If $D < 0$: Saddle point

   • If $D = 0$: Test inconclusive

Example: Find and classify critical points of $f(x, y) = x^2 + y^2 - 2x - 4y$

$$f_x = 2x - 2 = 0 \Rightarrow x = 1$$

$$f_y = 2y - 4 = 0 \Rightarrow y = 2$$

Critical point: $(1, 2)$

Second derivatives: $f_{xx} = 2$, $f_{yy} = 2$, $f_{xy} = 0$

$$D = (2)(2) - (0)^2 = 4 > 0$$

Since $f_{xx} = 2 > 0$, $(1, 2)$ is local minimum.

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13. Important Formulas Summary

13.1 Basic Differentiation Rules

• Power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$

• Product rule: $(fg)' = f'g + fg'$

• Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

• Chain rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$

13.2 Common Derivatives

• $\frac{d}{dx}(e^x) = e^x$

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$

• $\frac{d}{dx}(\sin x) = \cos x$

• $\frac{d}{dx}(\cos x) = -\sin x$

• $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$

13.3 Partial Derivatives Notation

• $f_x = \frac{\partial f}{\partial x}$

• $f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$

• Gradient: $\nabla f = \langle f_x, f_y \rangle$

13.4 Chain Rule for Partials

$$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t}$$

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14. Solved Examples

Example 1: Implicit Differentiation

Find $\frac{dy}{dx}$ for $x^2 + xy + y^2 = 3$

Solution:

Differentiate term by term:

$$2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$$

$$(x + 2y)\frac{dy}{dx} = -2x - y$$

$$\frac{dy}{dx} = \frac{-2x - y}{x + 2y}$$

Example 2: Logarithmic Differentiation

Find $\frac{dy}{dx}$ for $y = (\sin x)^x$

Solution:

Take $\ln$: $\ln y = x \ln(\sin x)$

Differentiate:

$$\frac{1}{y}\frac{dy}{dx} = \ln(\sin x) + x \cdot \frac{\cos x}{\sin x}$$

$$\frac{dy}{dx} = y\left[\ln(\sin x) + x\cot x\right] = (\sin x)^x[\ln(\sin x) + x\cot x]$$

Example 3: Partial Derivatives

Find all second partials of $f(x, y) = x^3y + e^{xy}$

Solution:

$$f_x = 3x^2y + ye^{xy}$$

$$f_y = x^3 + xe^{xy}$$

$$f_{xx} = \frac{\partial}{\partial x}(3x^2y + ye^{xy}) = 6xy + y^2e^{xy}$$

$$f_{yy} = \frac{\partial}{\partial y}(x^3 + xe^{xy}) = x^2e^{xy}$$

$$f_{xy} = \frac{\partial}{\partial y}(3x^2y + ye^{xy}) = 3x^2 + e^{xy} + xye^{xy}$$

$$f_{yx} = \frac{\partial}{\partial x}(x^3 + xe^{xy}) = 3x^2 + e^{xy} + xye^{xy}$$

Note $f_{xy} = f_{yx}$

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15. Common Mistakes and Exam Tips

15.1 Common Mistakes

1. Chain rule errors: Forgetting to multiply by derivative of inner function

2. Product/quotient rule: Misremembering formula signs

3. Partial derivatives: Forgetting which variable is held constant

4. Implicit differentiation: Forgetting to multiply by $\frac{dy}{dx}$ for $y$ terms

5. Logarithmic differentiation: Forgetting final step of multiplying by $y$

15.2 Problem-Solving Strategy

1. Identify type: Which rule(s) apply? (product, quotient, chain, implicit, etc.)

2. Proceed systematically: Write each step clearly

3. Check work: Verify derivative makes sense

4. Simplify: Final answer should be in simplest form

15.3 Quick Checks

1. Derivative of constant = 0

2. Chain rule: Always multiply by inside derivative

3. Product rule: $(fg)' = f'g + fg'$ (NOT $f'g'$)

4. Quotient rule: Denominator squared, minus sign in numerator

5. Partial derivatives: Only differentiate with respect to one variable at a time

This comprehensive theory covers all aspects of ordinary and partial differentiation with detailed explanations and examples, providing complete preparation for the entrance examination.