Basic Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Constant Rule | $\frac{d}{dx}(c) = 0$ | $\frac{d}{dx}(7) = 0$ |
| Power Rule | $\frac{d}{dx}(x^n) = nx^{n-1}$ | $\frac{d}{dx}(x^4) = 4x^3$ |
| Constant Multiple Rule | $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$ | $\frac{d}{dx}(3x^2) = 3 \cdot 2x = 6x$ |
| Sum/Difference Rule | $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ | $\frac{d}{dx}(x^3 + 5x) = 3x^2 + 5$ |
| Product Rule | $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ | $\frac{d}{dx}(x^2 \cdot \sin x) = 2x\sin x + x^2\cos x$ |
| Quotient Rule | $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}$ | $\frac{d}{dx}\left(\frac{x^2}{x+1}\right) = \frac{2x(x+1) – x^2 \cdot 1}{(x+1)^2} = \frac{2x^2+2x-x^2}{(x+1)^2} = \frac{x^2+2x}{(x+1)^2}$ |
| Chain Rule | $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ | $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)$ |
Common Function Derivatives
Basic Functions
| Function | Derivative |
|---|---|
| $c$ (constant) | $0$ |
| $x$ | $1$ |
| $x^n$ | $nx^{n-1}$ |
| $\sqrt{x}$ | $\frac{1}{2\sqrt{x}}$ |
| $\frac{1}{x}$ | $-\frac{1}{x^2}$ |
| $\frac{1}{x^n}$ | $-\frac{n}{x^{n+1}}$ |
Exponential and Logarithmic Functions
| Function | Derivative |
|---|---|
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln(a)$ |
| $\ln(x)$ | $\frac{1}{x}$ |
| $\log_a(x)$ | $\frac{1}{x \ln(a)}$ |
| $e^{kx}$ | $ke^{kx}$ |
| $\ln(ax)$ | $\frac{1}{x}$ |
Trigonometric Functions
| Function | Derivative |
|---|---|
| $\sin(x)$ | $\cos(x)$ |
| $\cos(x)$ | $-\sin(x)$ |
| $\tan(x)$ | $\sec^2(x)$ |
| $\cot(x)$ | $-\csc^2(x)$ |
| $\sec(x)$ | $\sec(x)\tan(x)$ |
| $\csc(x)$ | $-\csc(x)\cot(x)$ |
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 |
|---|---|
| $\sinh(x)$ | $\cosh(x)$ |
| $\cosh(x)$ | $\sinh(x)$ |
| $\tanh(x)$ | $\text{sech}^2(x)$ |
| $\coth(x)$ | $-\text{csch}^2(x)$ |
| $\text{sech}(x)$ | $-\text{sech}(x)\tanh(x)$ |
| $\text{csch}(x)$ | $-\text{csch}(x)\coth(x)$ |
Common Derivative Patterns with Chain Rule
Exponential Functions
| Function | Derivative |
|---|---|
| $e^{g(x)}$ | $e^{g(x)} \cdot g'(x)$ |
| $a^{g(x)}$ | $a^{g(x)} \cdot \ln(a) \cdot g'(x)$ |
| $\ln(g(x))$ | $\frac{g'(x)}{g(x)}$ |
| $\log_a(g(x))$ | $\frac{g'(x)}{g(x) \cdot \ln(a)}$ |
Trigonometric Functions
| Function | Derivative |
|---|---|
| $\sin(g(x))$ | $\cos(g(x)) \cdot g'(x)$ |
| $\cos(g(x))$ | $-\sin(g(x)) \cdot g'(x)$ |
| $\tan(g(x))$ | $\sec^2(g(x)) \cdot g'(x)$ |
| $\cot(g(x))$ | $-\csc^2(g(x)) \cdot g'(x)$ |
| $\sec(g(x))$ | $\sec(g(x))\tan(g(x)) \cdot g'(x)$ |
| $\csc(g(x))$ | $-\csc(g(x))\cot(g(x)) \cdot g'(x)$ |
Inverse Trigonometric Functions
| Function | Derivative |
|---|---|
| $\arcsin(g(x))$ | $\frac{g'(x)}{\sqrt{1-[g(x)]^2}}$ |
| $\arccos(g(x))$ | $-\frac{g'(x)}{\sqrt{1-[g(x)]^2}}$ |
| $\arctan(g(x))$ | $\frac{g'(x)}{1+[g(x)]^2}$ |
Powers and Roots
| Function | Derivative |
|---|---|
| $[g(x)]^n$ | $n[g(x)]^{n-1} \cdot g'(x)$ |
| $\sqrt{g(x)}$ | $\frac{g'(x)}{2\sqrt{g(x)}}$ |
| $\frac{1}{g(x)}$ | $-\frac{g'(x)}{[g(x)]^2}$ |
Common Composite Functions
| Function | Derivative |
|---|---|
| $(ax+b)^n$ | $an(ax+b)^{n-1}$ |
| $e^{ax+b}$ | $ae^{ax+b}$ |
| $\ln(ax+b)$ | $\frac{a}{ax+b}$ |
| $\sin(ax+b)$ | $a\cos(ax+b)$ |
| $\cos(ax+b)$ | $-a\sin(ax+b)$ |
| $\tan(ax+b)$ | $a\sec^2(ax+b)$ |
| $\arcsin(ax+b)$ | $\frac{a}{\sqrt{1-(ax+b)^2}}$ |
| $\arctan(ax+b)$ | $\frac{a}{1+(ax+b)^2}$ |
| $\sqrt{ax+b}$ | $\frac{a}{2\sqrt{ax+b}}$ |
Special Function Derivatives
| Function | Derivative |
|---|---|
| $x^x$ | $x^x(1+\ln x)$ |
| $\sin^2(x)$ | $2\sin(x)\cos(x)$ |
| $\cos^2(x)$ | $-2\cos(x)\sin(x)$ |
| $\sin(x)\cos(x)$ | $\cos^2(x)-\sin^2(x)$ |
| $e^x\sin(x)$ | $e^x\sin(x)+e^x\cos(x)$ |
| $\ln(x^n)$ | $\frac{n}{x}$ |
| $\ln(\sin x)$ | $\cot(x)$ |
| $\ln(\cos x)$ | $-\tan(x)$ |
| $\sin^{-1}(x)$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\sqrt{1-x^2}$ | $\frac{-x}{\sqrt{1-x^2}}$ |
| $\sqrt{a^2-x^2}$ | $\frac{-x}{\sqrt{a^2-x^2}}$ |
| $\sqrt{x^2+a^2}$ | $\frac{x}{\sqrt{x^2+a^2}}$ |
| $\sqrt{x^2-a^2}$ | $\frac{x}{\sqrt{x^2-a^2}}$ |
Derivatives Involving Multiple Rules
| Function | Derivative |
|---|---|
| $\sin^n(x)$ | $n\sin^{n-1}(x)\cos(x)$ |
| $\cos^n(x)$ | $-n\cos^{n-1}(x)\sin(x)$ |
| $\tan^n(x)$ | $n\tan^{n-1}(x)\sec^2(x)$ |
| $x^n\sin(x)$ | $nx^{n-1}\sin(x) + x^n\cos(x)$ |
| $x^n\cos(x)$ | $nx^{n-1}\cos(x) – x^n\sin(x)$ |
| $e^x\ln(x)$ | $e^x\ln(x) + \frac{e^x}{x}$ |
| $\frac{f(x)}{g(x)}$ | $\frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}$ |
| $\ln(f(x)g(x))$ | $\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)}$ |
| $f(x)^{g(x)}$ | $f(x)^{g(x)}\left[g'(x)\ln(f(x)) + \frac{g(x)f'(x)}{f(x)}\right]$ |
Derivative Shortcuts for Common Forms
| Function | Derivative |
|---|---|
| $u \cdot v$ | $u’ \cdot v + u \cdot v’$ |
| $\frac{u}{v}$ | $\frac{u’ \cdot v – u \cdot v’}{v^2}$ |
| $u \circ v$ | $(u’ \circ v) \cdot v’$ |
| $u^n$ | $n \cdot u^{n-1} \cdot u’$ |
| $e^u$ | $e^u \cdot u’$ |
| $\ln(u)$ | $\frac{u’}{u}$ |
| $\sin(u)$ | $\cos(u) \cdot u’$ |
| $\cos(u)$ | $-\sin(u) \cdot u’$ |
Derivatives of Parametric Equations
For x = f(t) and y = g(t):
$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{g'(t)}{f'(t)}$ (provided $f'(t) \neq 0$)
$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$
Derivatives of Implicit Functions
For an equation F(x,y) = 0:
- Differentiate both sides with respect to x
- Remember y is a function of x, so use the chain rule for terms with y
- Solve for $\frac{dy}{dx}$
Example: For $x^2 + y^2 = 25$
- Differentiate: $2x + 2y\frac{dy}{dx} = 0$
- Solve: $\frac{dy}{dx} = -\frac{x}{y}$
Logarithmic Differentiation
For functions with products, quotients, or variables in exponents:
- Take natural logarithm of both sides
- Use logarithm properties to simplify
- Differentiate both sides
- Solve for the derivative
Example: For $y = x^x$
- Take ln: $\ln(y) = \ln(x^x) = x\ln(x)$
- Differentiate: $\frac{1}{y}\frac{dy}{dx} = \ln(x) + 1$
- Solve: $\frac{dy}{dx} = y(\ln(x) + 1) = x^x(\ln(x) + 1)$
Derivative Applications and Formulas
Tangent and Normal Lines
- Slope of tangent line at point (a, f(a)): $m_{tangent} = f'(a)$
- Tangent line equation: $y – f(a) = f'(a)(x – a)$
- Slope of normal line: $m_{normal} = -\frac{1}{f'(a)}$
- Normal line equation: $y – f(a) = -\frac{1}{f'(a)}(x – a)$
Motion Formulas
For position function s(t):
- Velocity: $v(t) = s'(t)$
- Acceleration: $a(t) = v'(t) = s”(t)$
- Jerk: $j(t) = a'(t) = v”(t) = s”'(t)$
Related Rates
When variables x and y are related and changing with time:
- Use the chain rule: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$
Linear Approximation
Approximating f(x) near a point a:
- $f(x) \approx f(a) + f'(a)(x – a)$
Optimization
Finding extrema:
- Critical points: Set $f'(x) = 0$
- Second derivative test:
- If $f”(x) > 0$: Local minimum
- If $f”(x) < 0$: Local maximum
- If $f”(x) = 0$: Inconclusive (use first derivative test)
Mnemonic Devices for Remembering Derivatives
- Power Rule: “Bring the power down, subtract 1 from the power”
- Exponential e^x: “e^x stays the same”
- Sine and Cosine: “Sine becomes cosine, cosine becomes negative sine”
- Product Rule: “First times derivative of second plus second times derivative of first”
- Quotient Rule: “Bottom times derivative of top minus top times derivative of bottom, all over bottom squared”
- Chain Rule: “Outside function’s derivative evaluated at inside function, times inside function’s derivative”
This comprehensive cheatsheet covers the twelve most common derivative patterns and rules used in calculus. Keep it handy for quick reference during problem-solving or when working with complex derivative applications.