Derivative Rules

Trigonometric Derivatives

Exponential Derivatives

Logarithm Derivatives

Basic Rules

	law	formula	explanation
	Constant Rule	If $f(x) = c$ , then $f'(x) = 0$	The derivative of any constant is zero
	Constant Multiple Rule	If $g(x) = c \cdot f(x)$ , then $g'(x) = c \cdot f'(x)$	Constants can be factored out of derivatives
	Power Rule	If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$	Bring down the exponent and reduce the power by one
	Sum and Difference Rule	If $h(x) = f(x) \pm g(x)$ , then $h'(x) = f'(x) \pm g'(x)$	The derivative of a sum/difference is the sum/difference of derivatives
	Product Rule	If $h(x) = f(x)g(x)$ , then $h'(x) = f'(x)g(x) + f(x)g'(x)$	Derivative of first times second plus first times derivative of second
	Quotient Rule	If $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	Low d-high minus high d-low over low squared
	Chain Rule	If $h(x) = f(g(x))$ , then $h'(x) = f'(g(x))g'(x)$	Derivative of outside function times derivative of inside function

Trigonometric Derivatives

	law	formula	explanation
	Sine Derivative	If $f(x) = \sin(x)$ , then $f'(x) = \cos(x)$	The derivative of sine is cosine
	Cosine Derivative	If $f(x) = \cos(x)$ , then $f'(x) = -\sin(x)$	The derivative of cosine is negative sine
	Tangent Derivative	If $f(x) = \tan(x)$ , then $f'(x) = \sec^2(x)$	The derivative of tangent is secant squared
	Secant Derivative	If $f(x) = \sec(x)$ , then $f'(x) = \sec(x)\tan(x)$	The derivative of secant is secant times tangent
	Cotangent Derivative	If $f(x) = \cot(x)$ , then $f'(x) = -\csc^2(x)$	The derivative of cotangent is negative cosecant squared
	Cosecant Derivative	If $f(x) = \csc(x)$ , then $f'(x) = -\csc(x)\cot(x)$	The derivative of cosecant is negative cosecant times cotangent

Exponential Derivatives

	law	formula	explanation
	General Exponential	If $f(x) = a^x$ , then $f'(x) = \ln(a)a^x$	Derivative of exponential includes natural log of the base
	Natural Exponential	If $f(x) = e^x$ , then $f'(x) = e^x$	The derivative of e to the x is itself
	Composite General Exponential	If $f(x) = a^{g(x)}$ , then $f'(x) = \ln(a)a^{g(x)}g'(x)$	Use chain rule with general exponential derivative
	Composite Natural Exponential	If $f(x) = e^{g(x)}$ , then $f'(x) = e^{g(x)}g'(x)$	Use chain rule with natural exponential derivative

Logarithm Derivatives

	law	formula	explanation
	General Logarithm	If $f(x) = \log_a(x)$ , then $f'(x) = \frac{1}{\ln(a)x}$	Derivative of logarithm includes natural log of base in denominator
	Natural Logarithm	If $f(x) = \ln(x)$ , then $f'(x) = \frac{1}{x}$	The derivative of natural log is one over x
	Composite General Logarithm	If $f(x) = \log_a(g(x))$ , then $f'(x) = \frac{g'(x)}{\ln(a)g(x)}$	Use chain rule with general logarithm derivative
	Composite Natural Logarithm	If $f(x) = \ln(g(x))$ , then $f'(x) = \frac{g'(x)}{g(x)}$	Use chain rule with natural logarithm derivative