Introduction: Understanding Derivatives
A derivative represents the rate of change of a function with respect to its variable. It measures the instantaneous rate at which a quantity changes, providing the slope of the tangent line at any point on a function’s graph. Derivatives are fundamental to calculus, forming the basis for optimization, rate problems, approximations, and modeling dynamic systems in physics, economics, engineering, and numerous other disciplines.
Core Derivative Concepts
Definition of the Derivative
| Concept | Definition | Notation |
|---|---|---|
| Derivative at a point | $f'(a) = \lim_{h \to 0} \frac{f(a+h) – f(a)}{h}$ | $f'(a)$ or $\left.\frac{df}{dx}\right\vert_{x=a}$ |
| Derivative function | $f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$ | $f'(x)$, $\frac{df}{dx}$, or $\frac{d}{dx}f(x)$ |
| Alternate definition | $f'(x) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ | $\frac{dy}{dx}$ |
Notation Systems
| System | First Derivative | Second Derivative | Higher Derivatives |
|---|---|---|---|
| Lagrange | $f'(x)$ | $f”(x)$ | $f^{(n)}(x)$ |
| Leibniz | $\frac{df}{dx}$ or $\frac{d}{dx}f(x)$ | $\frac{d^2f}{dx^2}$ | $\frac{d^nf}{dx^n}$ |
| Newton | $\dot{y}$ | $\ddot{y}$ | No standard notation |
| Prime | $y’$ | $y”$ | $y^{(n)}$ |
Basic Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Constant Rule | $\frac{d}{dx}(c) = 0$ | $\frac{d}{dx}(5) = 0$ |
| Power Rule | $\frac{d}{dx}(x^n) = nx^{n-1}$ | $\frac{d}{dx}(x^3) = 3x^2$ |
| Constant Multiple Rule | $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$ | $\frac{d}{dx}(4x^2) = 4 \cdot 2x = 8x$ |
| Sum/Difference Rule | $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ | $\frac{d}{dx}(x^2 + 3x) = 2x + 3$ |
| Product Rule | $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ | $\frac{d}{dx}(x^2 \sin x) = 2x\sin x + x^2\cos x$ |
| Quotient Rule | $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}$ | $\frac{d}{dx}\left(\frac{x^2}{e^x}\right) = \frac{2xe^x – x^2e^x}{(e^x)^2}$ |
| Chain Rule | $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ | $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x$ |
Derivatives of Common Functions
Trigonometric Functions
| Function | Derivative |
|---|---|
| $\sin(x)$ | $\cos(x)$ |
| $\cos(x)$ | $-\sin(x)$ |
| $\tan(x)$ | $\sec^2(x)$ |
| $\cot(x)$ | $-\csc^2(x)$ |
| $\sec(x)$ | $\sec(x)\tan(x)$ |
| $\csc(x)$ | $-\csc(x)\cot(x)$ |
Exponential and Logarithmic Functions
| Function | Derivative |
|---|---|
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln(a)$ |
| $\ln(x)$ | $\frac{1}{x}$ |
| $\log_a(x)$ | $\frac{1}{x \ln(a)}$ |
Inverse Trigonometric Functions
| Function | Derivative |
|---|---|
| $\arcsin(x)$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\arccos(x)$ | $-\frac{1}{\sqrt{1-x^2}}$ |
| $\arctan(x)$ | $\frac{1}{1+x^2}$ |
| $\text{arccot}(x)$ | $-\frac{1}{1+x^2}$ |
| $\text{arcsec}(x)$ | $\frac{1}{ |
| $\text{arccsc}(x)$ | $-\frac{1}{ |
Hyperbolic Functions
| Function | Derivative |
|---|---|
| $\sinh(x)$ | $\cosh(x)$ |
| $\cosh(x)$ | $\sinh(x)$ |
| $\tanh(x)$ | $\text{sech}^2(x)$ |
| $\coth(x)$ | $-\text{csch}^2(x)$ |
| $\text{sech}(x)$ | $-\text{sech}(x)\tanh(x)$ |
| $\text{csch}(x)$ | $-\text{csch}(x)\coth(x)$ |
Advanced Differentiation Techniques
Implicit Differentiation
Used when a function is defined implicitly by an equation:
- Differentiate both sides of the equation with respect to x
- Remember to apply the chain rule for terms containing y by multiplying by dy/dx
- Solve for dy/dx
Example: For $x^2 + y^2 = 25$
- Differentiate: $2x + 2y\frac{dy}{dx} = 0$
- Solve: $\frac{dy}{dx} = -\frac{x}{y}$
Logarithmic Differentiation
Useful for functions with products, quotients, or variables in exponents:
- Take the natural logarithm of both sides
- Use logarithm properties to simplify
- Differentiate both sides with respect to x
- Solve for y’
Example: For $y = \frac{x^3\sqrt{x+1}}{(x-2)^2}$
- Take ln: $\ln(y) = \ln(x^3) + \ln(\sqrt{x+1}) – \ln((x-2)^2)$
- Simplify: $\ln(y) = 3\ln(x) + \frac{1}{2}\ln(x+1) – 2\ln(x-2)$
- Differentiate: $\frac{1}{y}\frac{dy}{dx} = \frac{3}{x} + \frac{1}{2} \cdot \frac{1}{x+1} \cdot 1 – 2 \cdot \frac{1}{x-2} \cdot 1$
- Multiply by y: $\frac{dy}{dx} = y\left(\frac{3}{x} + \frac{1}{2(x+1)} – \frac{2}{x-2}\right)$
Parametric Differentiation
For parametric equations x = x(t) and y = y(t):
- First derivative: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{y'(t)}{x'(t)}$, provided $x'(t) \neq 0$
- Second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$
Polar Curve Differentiation
For a curve r = f(θ):
- Cartesian coordinates: x = r cosθ, y = r sinθ
- Derivative: $\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta – r\sin\theta}$
Higher-Order Derivatives
| Derivative Order | Notation | Meaning |
|---|---|---|
| First derivative | $f'(x)$ or $\frac{df}{dx}$ | Rate of change of function |
| Second derivative | $f”(x)$ or $\frac{d^2f}{dx^2}$ | Rate of change of the first derivative (acceleration) |
| Third derivative | $f”'(x)$ or $\frac{d^3f}{dx^3}$ | Rate of change of the second derivative (jerk) |
| nth derivative | $f^{(n)}(x)$ or $\frac{d^nf}{dx^n}$ | Rate of change of the (n-1)th derivative |
Partial Derivatives
For functions of multiple variables f(x, y, z, …):
| Type | Notation | Meaning |
|---|---|---|
| First partial to x | $f_x$ or $\frac{\partial f}{\partial x}$ | Rate of change with respect to x, keeping other variables constant |
| First partial to y | $f_y$ or $\frac{\partial f}{\partial y}$ | Rate of change with respect to y, keeping other variables constant |
| Mixed partial | $f_{xy}$ or $\frac{\partial^2 f}{\partial y \partial x}$ | Partial of a partial (order matters unless continuous) |
Applications of Derivatives
Related Rates
Problems where rates of change of related variables are given:
- Identify variables and how they’re related
- Write equation connecting variables
- Differentiate with respect to time
- Substitute known values to find unknown rate
Optimization Problems
Steps to find extrema:
- Identify the function to optimize
- Find critical points by setting f'(x) = 0
- Use second derivative test:
- If f”(x) > 0: Local minimum
- If f”(x) < 0: Local maximum
- If f”(x) = 0: Inconclusive (use first derivative test)
- Check endpoints if domain is restricted
- Determine global extrema from all critical points and endpoints
Mean Value Theorem
If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists at least one point c in (a,b) such that:
$f'(c) = \frac{f(b) – f(a)}{b – a}$
Linear Approximation
Approximating f(x) near a point a:
$f(x) \approx f(a) + f'(a)(x – a)$
L’Hôpital’s Rule
For limits of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$:
$\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$
(Apply repeatedly if needed)
Common Derivatives Challenges and Solutions
Challenge: Complex Composite Functions
Example: $f(x) = \sin(e^{x^2 + 3x})$
Solution: Apply chain rule multiple times, working from outside in:
- Let u = e^{x^2 + 3x}, so f(x) = sin(u)
- f'(x) = cos(u) · u’
- u’ = e^{x^2 + 3x} · (x^2 + 3x)’
- (x^2 + 3x)’ = 2x + 3
- Therefore, f'(x) = cos(e^{x^2 + 3x}) · e^{x^2 + 3x} · (2x + 3)
Challenge: Functions Not Explicitly Solved for y
Example: $x^3 + y^3 = 6xy$
Solution: Use implicit differentiation:
- Differentiate both sides: 3x^2 + 3y^2 · y’ = 6y + 6x · y’
- Group terms with y’: 3y^2 · y’ – 6x · y’ = 6y – 3x^2
- Factor out y’: (3y^2 – 6x) · y’ = 6y – 3x^2
- Solve: $y’ = \frac{6y – 3x^2}{3y^2 – 6x}$
Challenge: Products of Many Functions
Example: $f(x) = x^2 \sin(x) e^x \ln(x)$
Solution: Use logarithmic differentiation:
- ln(f(x)) = ln(x^2) + ln(sin(x)) + ln(e^x) + ln(ln(x))
- ln(f(x)) = 2ln(x) + ln(sin(x)) + x + ln(ln(x))
- Differentiate: $\frac{f'(x)}{f(x)} = \frac{2}{x} + \frac{\cos(x)}{\sin(x)} + 1 + \frac{1}{\ln(x)} \cdot \frac{1}{x}$
- Multiply by f(x): $f'(x) = f(x) \left(\frac{2}{x} + \cot(x) + 1 + \frac{1}{x\ln(x)}\right)$
Best Practices and Tips
- Before differentiating, simplify the expression if possible.
- Identify the appropriate rule based on the function’s structure:
- Sum/difference → Sum/Difference Rule
- Product → Product Rule
- Quotient → Quotient Rule
- Composition → Chain Rule
- For complex expressions, break them down into simpler parts.
- When in doubt, use logarithmic differentiation for complicated products/quotients.
- Double-check your work by testing with simple values.
- Remember that differentiation is linear: d/dx[af(x) + bg(x)] = a·f'(x) + b·g'(x)
- Create a substitution (u = g(x)) for complex chain rule applications.
- Memorize the basic derivatives to save time and reduce errors.
- Be careful with signs when applying the chain rule or quotient rule.
- For piecewise functions, differentiate each piece separately and check continuity.
Derivative Shortcut Table
| Function | Derivative |
|---|---|
| $x^n$ | $nx^{n-1}$ |
| $\sqrt{x}$ | $\frac{1}{2\sqrt{x}}$ |
| $\frac{1}{x}$ | $-\frac{1}{x^2}$ |
| $\frac{1}{x^n}$ | $-\frac{n}{x^{n+1}}$ |
| $e^{kx}$ | $ke^{kx}$ |
| $\ln(kx)$ | $\frac{1}{x}$ |
| $\sin(kx)$ | $k\cos(kx)$ |
| $\cos(kx)$ | $-k\sin(kx)$ |
| $\tan(kx)$ | $k\sec^2(kx)$ |
| $a^x$ | $a^x\ln(a)$ |
| $\sin^2(x)$ | $2\sin(x)\cos(x)$ |
| $\cos^2(x)$ | $-2\cos(x)\sin(x)$ |
| $\sqrt{1-x^2}$ | $\frac{-x}{\sqrt{1-x^2}}$ |
| $\sqrt{a^2-x^2}$ | $\frac{-x}{\sqrt{a^2-x^2}}$ |
| $\sqrt{x^2+a^2}$ | $\frac{x}{\sqrt{x^2+a^2}}$ |
| $\sqrt{x^2-a^2}$ | $\frac{x}{\sqrt{x^2-a^2}}$ |
Resources for Further Learning
Textbooks and Reference Materials
- “Calculus” by James Stewart
- “Calculus: Early Transcendentals” by Jon Rogawski
- “Calculus Made Easy” by Silvanus P. Thompson
Online Resources
- Khan Academy (interactive calculus courses)
- Paul’s Online Math Notes (comprehensive derivative guides)
- MIT OpenCourseWare (calculus lectures)
- 3Blue1Brown (YouTube channel for visual calculus)
Practice Tools
- Symbolab (step-by-step derivative calculator)
- Wolfram Alpha (derivative calculator with explanations)
- Desmos (graphing calculator to visualize derivatives)
This cheatsheet provides a comprehensive reference for derivatives in calculus, covering basic rules, common functions, advanced techniques, and practical applications. Use it as a quick reference during problem-solving or to reinforce your understanding of key differentiation concepts.