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Basic Functions and Their Derivatives

Standard Functions​

Function TypeFunctionDerivative
Constant Functionf(x)=cf(x) = cf′(x)=0f'(x) = 0
Linear Functionf(x)=mx+bf(x) = mx + bf′(x)=mf'(x) = m
Quadratic Functionf(x)=ax2+bx+cf(x) = ax^2 + bx + cf′(x)=2ax+bf'(x) = 2ax + b
Exponential Function (base e)f(x)=exf(x) = e^xf′(x)=exf'(x) = e^x
Logarithmic Function (base e)f(x)=ln⁡(x)f(x) = \ln(x)f′(x)=1xf'(x) = \frac{1}{x}
Exponential Function (general base)f(x)=axf(x) = a^xf′(x)=axln⁡(a)f'(x) = a^x \ln(a)
Logarithmic Function (general base)f(x)=log⁡a(x)f(x) = \log_a(x)f′(x)=1xln⁡(a)f'(x) = \frac{1}{x \ln(a)}
Derivative of a Squareddx[f(x)]2\frac{d}{dx} [f(x)]^22f(x)f′(x)2 f(x) f'(x)
Derivative of a Reciprocalddx(1g(x))\frac{d}{dx} \left(\frac{1}{g(x)}\right)−g′(x)[g(x)]2-\frac{g'(x)}{[g(x)]^2}

Trigonometric Functions​

Function TypeFunctionDerivative
Sine Functionf(x)=sin⁡(x)f(x) = \sin(x)f′(x)=cos⁡(x)f'(x) = \cos(x)
Cosine Functionf(x)=cos⁡(x)f(x) = \cos(x)f′(x)=−sin⁡(x)f'(x) = -\sin(x)
Tangent Functionf(x)=tan⁡(x)f(x) = \tan(x)f′(x)=sec⁡2(x)f'(x) = \sec^2(x)
Cotangent Functionf(x)=cot⁡(x)f(x) = \cot(x)f′(x)=−csc⁡2(x)f'(x) = -\csc^2(x)
Secant Functionf(x)=sec⁡(x)f(x) = \sec(x)f′(x)=sec⁡(x)tan⁡(x)f'(x) = \sec(x)\tan(x)
Cosecant Functionf(x)=csc⁡(x)f(x) = \csc(x)f′(x)=−csc⁡(x)cot⁡(x)f'(x) = -\csc(x)\cot(x)
Inverse Sine Functionf(x)=sin⁡−1(x)f(x) = \sin^{-1}(x)f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1-x^2}}
Inverse Cosine Functionf(x)=cos⁡−1(x)f(x) = \cos^{-1}(x)f′(x)=−11−x2f'(x) = -\frac{1}{\sqrt{1-x^2}}
Inverse Tangent Functionf(x)=tan⁡−1(x)f(x) = \tan^{-1}(x)f′(x)=11+x2f'(x) = \frac{1}{1+x^2}

Chain Rule Variants​

Chain Rule Derivative
ddxf(x)n=nf(x)n−1⋅f′(x)\frac{d}{dx} f(x)^n = n f(x)^{n-1} \cdot f'(x)
ddxef(x)=ef(x)⋅f′(x)\frac{d}{dx} e^{f(x)} = e^{f(x)} \cdot f'(x)
ddxln⁡(f(x))=f′(x)f(x)\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}
ddxsin⁡(f(x))=cos⁡(f(x))⋅f′(x)\frac{d}{dx} \sin(f(x)) = \cos(f(x)) \cdot f'(x)
ddxcos⁡(f(x))=−sin⁡(f(x))⋅f′(x)\frac{d}{dx} \cos(f(x)) = -\sin(f(x)) \cdot f'(x)
ddxtan⁡(f(x))=sec⁡2(f(x))⋅f′(x)\frac{d}{dx} \tan(f(x)) = \sec^2(f(x)) \cdot f'(x)
ddxsec⁡(f(x))=sec⁡(f(x))tan⁡(f(x))⋅f′(x)\frac{d}{dx} \sec(f(x)) = \sec(f(x)) \tan(f(x)) \cdot f'(x)
ddxtan⁡−1(f(x))=f′(x)1+f(x)2\frac{d}{dx} \tan^{-1}(f(x)) = \frac{f'(x)}{1 + f(x)^2}

References​