Differentiation Formulas Guide: Basic to Intermediate with Examples

Complete Differentiation Formulas: Basic to Intermediate

This guide covers every essential differentiation formula from fundamentals to intermediate techniques, with worked examples and practice problems.

1. Basic Differentiation Rules

1.1 Fundamental Rules

Rule	Formula
Constant Rule	$\displaystyle \frac{d}{dx}(c) = 0$
Power Rule	$\displaystyle \frac{d}{dx}(x^n) = nx^{n-1}$
Constant Multiple	$\displaystyle \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$
Sum/Difference	$\displaystyle \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

1.2 Example: Power Rule

Problem: Find $\displaystyle \frac{d}{dx}(5x^4)$

Solution:

\frac{d}{dx}(5x^4) = 5 \cdot 4x^{4-1} = 20x^3

2. Product Rule, Quotient Rule, and Chain Rule

2.1 Product Rule

If $y = u \cdot v$ where $u$ and $v$ are functions of $x$ :

\frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}

Or: $(uv)' = u'v + uv'$

2.2 Quotient Rule

If $y = \frac{u}{v}$ where $v \neq 0$ :

\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}

Or: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

2.3 Chain Rule

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

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

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

2.4 Example 1: Product Rule

Problem: Differentiate $y = x^2 \sin x$

Solution: Let $u = x^2$ and $v = \sin x$ .

$u' = 2x$
$v' = \cos x$

\frac{dy}{dx} = x^2 \cdot \cos x + \sin x \cdot 2x = x^2\cos x + 2x\sin x

2.5 Example 2: Quotient Rule

Problem: Differentiate $y = \frac{3x+1}{x^2+1}$

Solution: Let $u = 3x+1$ and $v = x^2+1$ .

$u' = 3$
$v' = 2x$

\frac{dy}{dx} = \frac{(x^2+1)(3) - (3x+1)(2x)}{(x^2+1)^2} = \frac{3x^2+3 - 6x^2-2x}{(x^2+1)^2} = \frac{-3x^2-2x+3}{(x^2+1)^2}

2.6 Example 3: Chain Rule

Problem: Differentiate $y = (3x^2 + 1)^5$

Solution: Let $u = 3x^2 + 1$ , so $y = u^5$ .

$\frac{dy}{du} = 5u^4$
$\frac{du}{dx} = 6x$

\frac{dy}{dx} = 5u^4 \cdot 6x = 5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4

3. Standard Derivatives

3.1 Algebraic and Power Functions

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

\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}

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

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

3.2 Exponential and Logarithmic Functions

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

\frac{d}{dx}(a^x) = a^x \ln a

\frac{d}{dx}(\ln x) = \frac{1}{x}

\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}

\frac{d}{dx}(e^{ax}) = ae^{ax}

\frac{d}{dx}(\ln(ax+b)) = \frac{a}{ax+b}

3.3 Trigonometric Functions

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

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

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

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

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

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

3.4 Inverse Trigonometric Functions

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

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

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

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

\frac{d}{dx}(\sec^{-1} x) = \frac{1}{x\sqrt{x^2-1}}

\frac{d}{dx}(\csc^{-1} x) = \frac{-1}{x\sqrt{x^2-1}}

3.5 Generalized Inverse Trigonometric Forms

\frac{d}{dx}(\sin^{-1}(ax+b)) = \frac{a}{\sqrt{1-(ax+b)^2}}

\frac{d}{dx}(\tan^{-1}(ax+b)) = \frac{a}{1+(ax+b)^2}

3.6 Hyperbolic Functions

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

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

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

\frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{x^2+1}}

\frac{d}{dx}(\cosh^{-1} x) = \frac{1}{\sqrt{x^2-1}}

\frac{d}{dx}(\tanh^{-1} x) = \frac{1}{1-x^2} \quad (|x| < 1)

3.7 Example: Combined Rules

Problem: Differentiate $y = e^{x^2} \ln x$

Solution: Use product rule: $u = e^{x^2}$ , $v = \ln x$

For $u = e^{x^2}$ , use chain rule:

\frac{du}{dx} = e^{x^2} \cdot 2x = 2xe^{x^2}

For $v = \ln x$ :

\frac{dv}{dx} = \frac{1}{x}

Apply product rule:

\frac{dy}{dx} = e^{x^2} \cdot \frac{1}{x} + \ln x \cdot 2xe^{x^2} = \frac{e^{x^2}}{x} + 2xe^{x^2}\ln x

Factor:

= \frac{e^{x^2}(1 + 2x^2\ln x)}{x}

4. Implicit Differentiation

4.1 Method

When $y$ is defined implicitly by an equation $F(x, y) = 0$ :

Differentiate both sides with respect to $x$
Remember that $\frac{d}{dx}(y^n) = ny^{n-1}\frac{dy}{dx}$ (chain rule)

4.2 Example 1: Circle

Problem: Find $\frac{dy}{dx}$ if $x^2 + y^2 = 25$

Solution: Differentiate both sides with respect to $x$ :

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

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

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

4.3 Example 2: Product Within Implicit

Problem: Find $\frac{dy}{dx}$ if $x^3 + xy + y^3 = 7$

Solution: Differentiate term by term:

3x^2 + \left(1 \cdot y + x \cdot \frac{dy}{dx}\right) + 3y^2\frac{dy}{dx} = 0

Group terms with $\frac{dy}{dx}$ :

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

Factor:

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

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

5. Logarithmic Differentiation

5.1 When to Use

Useful for:

Functions of the form $y = [f(x)]^{g(x)}$
Products/quotients with many factors
Complex exponential forms

5.2 Method

Take natural logarithm of both sides: $\ln y = \ln(f(x))$
Use logarithm properties to simplify
Differentiate implicitly with respect to $x$
Solve for $\frac{dy}{dx}$

5.3 Example 1: Variable Power

Problem: Differentiate $y = x^x$ for $x > 0$

Solution: Take logarithm:

\ln y = \ln(x^x) = x\ln x

Differentiate both sides:

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

Multiply by $y$ :

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

5.4 Example 2: Complex Product/Quotient

Problem: Differentiate $y = \frac{(x+1)^3(x-2)^2}{(x+3)^4}$

Solution: Take logarithm:

\ln y = 3\ln(x+1) + 2\ln(x-2) - 4\ln(x+3)

Differentiate:

\frac{1}{y}\frac{dy}{dx} = \frac{3}{x+1} + \frac{2}{x-2} - \frac{4}{x+3}

Multiply by $y$ :

\frac{dy}{dx} = \frac{(x+1)^3(x-2)^2}{(x+3)^4}\left[\frac{3}{x+1} + \frac{2}{x-2} - \frac{4}{x+3}\right]

6. Parametric Differentiation

6.1 Method

If $x = f(t)$ and $y = g(t)$ , then:

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}

Second derivative:

\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

6.2 Example

Problem: Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ if and

Solution: First derivatives with respect to $t$ :

\frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2 - 3

First derivative:

\frac{dy}{dx} = \frac{3t^2 - 3}{2t} = \frac{3(t^2-1)}{2t}

Second derivative:

\frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{3t^2-3}{2t}\right) = \frac{6t \cdot 2t - (3t^2-3) \cdot 2}{4t^2} = \frac{12t^2 - 6t^2 + 6}{4t^2} = \frac{6t^2+6}{4t^2} = \frac{3(t^2+1)}{2t^2}

\frac{d^2y}{dx^2} = \frac{3(t^2+1)}{2t^2} \cdot \frac{1}{2t} = \frac{3(t^2+1)}{4t^3}

7. Higher-Order Derivatives

7.1 Notation

f'(x) = \frac{dy}{dx}, \quad f''(x) = \frac{d^2y}{dx^2}, \quad f'''(x) = \frac{d^3y}{dx^3}, \quad f^{(n)}(x) = \frac{d^ny}{dx^n}

7.2 Standard $n$ -th Derivatives

\frac{d^n}{dx^n}(e^{ax}) = a^n e^{ax}

\frac{d^n}{dx^n}(\sin(ax+b)) = a^n \sin\left(ax+b+\frac{n\pi}{2}\right)

\frac{d^n}{dx^n}(\cos(ax+b)) = a^n \cos\left(ax+b+\frac{n\pi}{2}\right)

\frac{d^n}{dx^n}(\ln(x)) = \frac{(-1)^{n-1}(n-1)!}{x^n}

\frac{d^n}{dx^n}\left(\frac{1}{ax+b}\right) = \frac{(-1)^n n! \, a^n}{(ax+b)^{n+1}}

7.3 Leibniz Rule for Product

\frac{d^n}{dx^n}(uv) = \sum_{r=0}^{n} \binom{n}{r} u^{(n-r)} v^{(r)}

7.4 Example: Higher-Order Derivative

Problem: Find $\frac{d^3y}{dx^3}$ if $y = x^{4} + 2 x^{3} -$

Solution:

\frac{dy}{dx} = 4x^3 + 6x^2 - 1

\frac{d^2y}{dx^2} = 12x^2 + 12x

\frac{d^3y}{dx^3} = 24x + 12

8. Applications of Derivatives

8.1 Rate of Change

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

Example: The radius of a circle increases at $2$ cm/s. How fast is the area increasing when $r = 5$ cm?

Solution: $A = \pi r^2$

\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi(5)(2) = 20\pi \text{ cm}^2/\text{s}

8.2 Tangent and Normal Lines

For $y = f(x)$ at point $(x_1, y_1)$ :

Slope of tangent: $m = f'(x_1)$

Equation of tangent:

y - y_1 = f'(x_1)(x - x_1)

Slope of normal: $m_{\perp} = -\frac{1}{f'(x_1)}$ (if )

Equation of normal:

y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)

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

Solution:

\frac{dy}{dx} = 2x \implies m = 4

Tangent: $y - 4 = 4(x - 2) \implies y = 4x - 4$

8.3 Increasing and Decreasing Functions

$f'(x) > 0$ on interval $\Rightarrow$ $f$ is increasing
$f'(x) < 0$ on interval is

8.4 Maxima and Minima (First Derivative Test)

At critical point $c$ where $f'(c) = 0$ :

If $f'$ changes from $+$ to $-$ : local maximum
If $f'$ changes from to :

8.5 Maxima and Minima (Second Derivative Test)

At critical point $c$ :

$f''(c) < 0$ : local maximum
$f''(c) > 0$ :

8.6 Example: Optimization

Problem: Find the dimensions of a rectangle with perimeter $40$ m that maximizes area.

Solution: Let length $= l$ , width $= w$ . Constraint: $2l + 2w = 40 \Rightarrow l + w = 20 \Rightarrow w = 20 - l$

Area: $A = l(20-l) = 20l - l^2$

\frac{dA}{dl} = 20 - 2l = 0 \implies l = 10

\frac{d^2A}{dl^2} = -2 < 0 \implies \text{maximum}

So $l = 10$ m, $w = 10$ m. Maximum area is a square of $100$ m².

9. Rolle's Theorem and Mean Value Theorem

9.1 Rolle's Theorem

If $f(x)$ is:

Continuous on $[a, b]$
Differentiable on $(a, b)$
$f(a) = f(b)$

Then there exists at least one $c \in (a, b)$ such that:

f'(c) = 0

9.2 Lagrange's Mean Value Theorem (LMVT)

If $f(x)$ is:

Continuous on $[a, b]$
Differentiable on $(a, b)$

Then there exists at least one $c \in (a, b)$ such that:

f'(c) = \frac{f(b) - f(a)}{b - a}

10. Solved Examples (Mixed Practice)

Example 1: Chain Rule Nested

Problem: Differentiate $y = \sin^3(2x+1)$

Solution: Let $y = u^3$ where $u = \sin v$ and $v = 2x+1$ .

\frac{dy}{dx} = 3u^2 \cdot \cos v \cdot 2 = 3\sin^2(2x+1) \cdot \cos(2x+1) \cdot 2

= 6\sin^2(2x+1)\cos(2x+1)

Example 2: Implicit with Trigonometric

Problem: Find $\frac{dy}{dx}$ if $\sin y + \cos x = xy$

Solution: Differentiate:

\cos y \cdot \frac{dy}{dx} - \sin x = y + x\frac{dy}{dx}

Group:

\frac{dy}{dx}(\cos y - x) = y + \sin x

\frac{dy}{dx} = \frac{y + \sin x}{\cos y - x}

Example 3: Related Rates

Problem: A ladder $5$ m long rests against a wall. The bottom slides away at $2$ m/s. How fast is the top sliding down when the bottom is $3$ m from the wall?

Solution: Let $x$ = distance from wall, $y$ = height on wall. $x^2 + y^2 = 25$

When $x = 3$ : $y = 4$ .

Differentiate with respect to $t$ :

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

2(3)(2) + 2(4)\frac{dy}{dt} = 0

12 + 8\frac{dy}{dt} = 0 \implies \frac{dy}{dt} = -\frac{12}{8} = -1.5 \text{ m/s}

The negative indicates the top is sliding down at $1.5$ m/s.

11. Practice Problems

Level 1 — Basic Differentiation

$\displaystyle \frac{d}{dx}(7x^5 - 3x^2 + 2x - 1)$

Level 2 — Product, Quotient, Chain

$\displaystyle \frac{d}{dx}(x^2 e^x)$
$\frac{d}{d x}$

Level 3 — Implicit, Logarithmic, Parametric

Find $\frac{dy}{dx}$ if $x^2 + xy + y^2 = 7$

Level 4 — Applications

Find the equation of the tangent to $y = x^3 - 2x$ at $x = 2$
Find local maxima/minima of $f(x) = x^3 - 3x^2 + 4$

12. Answers to Practice Problems

Problem	Answer
1	$35x^4 - 6x + 2$
2	$-\frac{6}{x^4}$

Quick Reference Formula Sheet

Function	Derivative
$x^n$	$nx^{n-1}$
$e^x$

Rule	Formula
Product	$(uv)' = u'v + uv'$

End of Differentiation Formulas Guide

Complete Differentiation Formulas: Basic to Intermediate

1. Basic Differentiation Rules

1.1 Fundamental Rules

1.2 Example: Power Rule

2. Product Rule, Quotient Rule, and Chain Rule

2.1 Product Rule

2.2 Quotient Rule

2.3 Chain Rule

2.4 Example 1: Product Rule

2.5 Example 2: Quotient Rule

2.6 Example 3: Chain Rule

3. Standard Derivatives

3.1 Algebraic and Power Functions

3.2 Exponential and Logarithmic Functions

3.3 Trigonometric Functions

3.4 Inverse Trigonometric Functions

3.5 Generalized Inverse Trigonometric Forms

3.6 Hyperbolic Functions

3.7 Example: Combined Rules

4. Implicit Differentiation

4.1 Method

4.2 Example 1: Circle

4.3 Example 2: Product Within Implicit

5. Logarithmic Differentiation

5.1 When to Use

5.2 Method

5.3 Example 1: Variable Power

5.4 Example 2: Complex Product/Quotient

6. Parametric Differentiation

6.1 Method

6.2 Example

7. Higher-Order Derivatives

7.1 Notation

7.2 Standard nnn-th Derivatives

7.3 Leibniz Rule for Product

7.4 Example: Higher-Order Derivative

8. Applications of Derivatives

8.1 Rate of Change

8.2 Tangent and Normal Lines

8.3 Increasing and Decreasing Functions

8.4 Maxima and Minima (First Derivative Test)

8.5 Maxima and Minima (Second Derivative Test)

8.6 Example: Optimization

9. Rolle's Theorem and Mean Value Theorem

9.1 Rolle's Theorem

9.2 Lagrange's Mean Value Theorem (LMVT)

10. Solved Examples (Mixed Practice)

Example 1: Chain Rule Nested

Example 2: Implicit with Trigonometric

Example 3: Related Rates

11. Practice Problems

Level 1 — Basic Differentiation

Level 2 — Product, Quotient, Chain

Level 3 — Implicit, Logarithmic, Parametric

Level 4 — Applications

12. Answers to Practice Problems

Quick Reference Formula Sheet

7.2 Standard $n$ -th Derivatives