Differentiation Formulas

Basic Properties/Formulas/Rules

• \dfrac{d}{dx} [cf(x)] = cf'(x) \\
• \dfrac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \\
• \dfrac{d}{dx}[f(x) \space \pm \space g(x)] = f'(x) \space \pm \space g'(x) \\
• \dfrac{d}{dx} \bigg[\dfrac{f(x)}{g(x)}\bigg]= \dfrac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

Chain Rule

• \dfrac{d}{dx}[f(g(x))] = f'(g(x)) g'(x) \\
• \dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}

Common Derivatives

Polynomials

• \dfrac{d}{dx}[c] = 0 \\
• \dfrac{d}{dx}[cx] = c \\
• \dfrac{d}{dx}[cn^n] =ncx^{n-1} \\
• \dfrac{d}{dx}[x] = 1 \\
• \dfrac{d}{dx}[x^n] = nx^{n-1}

Trigonometric Functions

• \dfrac{d}{dx}[\sin(x)] = \cos(x) \\
• \dfrac{d}{dx}[\tan(x)] = \sec^2(x) \\
• \dfrac{d}{dx}[\sec(x)] = \sec(x) \tan(x) \\
• \dfrac{d}{dx}[\cos(x)] = - \sin(x) \\
• \dfrac{d}{dx}[\cot(x)] = -\csc^2(x) \\
• \dfrac{d}{dx}[\csc(x)] = -\csc(x) \cot (x)

Inverse Trigonometric Functions

• \dfrac{d}{dx}[\sin^{-1}(x)] = \dfrac{1}{\sqrt{1-x^2}} \\
• \dfrac{d}{dx}[\tan^{-1}(x)] = \dfrac{1}{1+x^2} \\
• \dfrac{d}{dx}[\sec^{-1}(x)] = \dfrac{1}{x \sqrt{x^2 -1}} \\
• \dfrac{d}{dx}[\cos^{-1}(x)] = -\dfrac{1}{\sqrt{1-x^2}} \\
• \dfrac{d}{dx}[\cot^{-1}(x)] = -\dfrac{1}{1+x^2} \\
• \dfrac{d}{dx}[\csc^{-1}(x)] = -\dfrac{1}{x\sqrt{x^2-1}}

Exponential and Logarithmic Functions

• \dfrac{d}{dx}[e^2] = e^x \\
• \dfrac{d}{dx}[\text{ln}(x)] =\dfrac{1}{x} \\
• \dfrac{d}{dx}[a^x] = a^x\text{ ln}(a) \\
• \dfrac{x}{dx}[log_a(x)] = \dfrac{1}{x \text{ ln}(a)}