Differentiation Rules

A complete set of vocabulary flashcards covering basic, exponential, logarithmic, trigonometric, inverse trigonometric, and hyperbolic differentiation formulas.

Constant Rule

The rule stating that the derivative of any constant $c$ is zero: $\frac{d}{dx}(c) = 0$

Constant Multiple Rule

The rule stating that a constant factor can be factored out of a derivative: $\frac{d}{dx}[cf(x)] = cf'(x)$

Sum Rule

The rule stating that the derivative of a sum of functions equals the sum of their individual derivatives: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

Difference Rule

The rule stating that the derivative of a difference of functions equals the difference of their individual derivatives: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$

Product Rule

The formula for computing the derivative of a product of two differentiable functions: $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$

Quotient Rule

The formula for differentiating the ratio of two differentiable functions: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$

Chain Rule

The formula for finding the derivative of a composite function: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

Power Rule

The differentiation formula for a power of $x$: $\frac{d}{dx}(x^n) = nx^{n-1}$

Derivative of $e^x$

The derivative of the natural exponential function: $\frac{d}{dx}(e^x) = e^x$

Derivative of $b^x$

The derivative of a general exponential function with base $b$: $\frac{d}{dx}(b^x) = b^x \ln(b)$

Derivative of $\ln|x|$

The derivative of the natural logarithm of the absolute value of $x$: $\frac{d}{dx}\ln|x| = \frac{1}{x}$

Derivative of $\log_b(x)$

The derivative of a general logarithm function with base $b$: $\frac{d}{dx}(\log_b(x)) = \frac{1}{x \ln(b)}$

Derivative of $\sin(x)$

The derivative of the sine function: $\frac{d}{dx}(\sin(x)) = \cos(x)$

Derivative of $\cos(x)$

The derivative of the cosine function: $\frac{d}{dx}(\cos(x)) = -\sin(x)$

Derivative of $\tan(x)$

The derivative of the tangent function: $\frac{d}{dx}(\tan(x)) = \sec^2(x)$

Derivative of $\csc(x)$

The derivative of the cosecant function: $\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$

Derivative of $\sec(x)$

The derivative of the secant function: $\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$

Derivative of $\cot(x)$

The derivative of the cotangent function: $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$

Derivative of $\sin^{-1}(x)$

The derivative of the inverse sine function: $\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}}$

Derivative of $\cos^{-1}(x)$

The derivative of the inverse cosine function: $\frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1 - x^2}}$

Derivative of $\tan^{-1}(x)$

The derivative of the inverse tangent function: $\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1 + x^2}$

Derivative of $\csc^{-1}(x)$

The derivative of the inverse cosecant function: $\frac{d}{dx}(\csc^{-1}(x)) = -\frac{1}{x\sqrt{x^2 - 1}}$

Derivative of $\sec^{-1}(x)$

The derivative of the inverse secant function: $\frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{x\sqrt{x^2 - 1}}$

Derivative of $\cot^{-1}(x)$

The derivative of the inverse cotangent function: $\frac{d}{dx}(\cot^{-1}(x)) = -\frac{1}{1 + x^2}$

Derivative of $\sinh(x)$

The derivative of the hyperbolic sine function: $\frac{d}{dx}(\sinh(x)) = \cosh(x)$

Derivative of $\cosh(x)$

The derivative of the hyperbolic cosine function: $\frac{d}{dx}(\cosh(x)) = \sinh(x)$

Derivative of $\tanh(x)$

The derivative of the hyperbolic tangent function: $\frac{d}{dx}(\tanh(x)) = \text{sech}^2(x)$

Derivative of $\text{csch}(x)$

The derivative of the hyperbolic cosecant function: $\frac{d}{dx}(\text{csch}(x)) = -\text{csch}(x)\coth(x)$

Derivative of $\text{sech}(x)$

The derivative of the hyperbolic secant function: $\frac{d}{dx}(\text{sech}(x)) = -\text{sech}(x)\tanh(x)$

Derivative of $\coth(x)$

The derivative of the hyperbolic cotangent function: $\frac{d}{dx}(\coth(x)) = -\text{csch}^2(x)$