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Table of Derivatives

General Formulas

  1. ddx(c)=0\frac{d}{dx}(c)=0
  2. ddx[cf(x)]=cf′(x)\frac{d}{dx}[cf(x)]=cf^{\prime}(x)
  3. ddx(xn)=nxn−1, for real numbers n\frac{d}{dx}(x^n)=nx^{n-1}\text{, for real numbers }n
  4. ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x)+g(x)]=f^{\prime}(x)+g^{\prime}(x)
  5. ddx[f(x)−g(x)]=f′(x)−g′(x)\frac{d}{dx}[f(x)-g(x)]=f^{\prime}(x)-g^{\prime}(x)
  6. ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x)g(x)]=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)
  7. ddx[f(x)g(x)]=g(x)f′(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f^{\prime}(x)-f(x)g^{\prime}(x)}{[g(x)]^2}
  8. ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))]=f^{\prime}(g(x))\cdot g^{\prime}(x)

Trigonometric Functions

  1. ddx[sin⁡(x)]=cos⁡(x)\frac{d}{dx}[\sin(x)]=\cos(x)
  2. ddx[cos⁡(x)]=−sin⁡(x)\frac{d}{dx}[\cos(x)]=-\sin(x)
  3. ddx[tan⁡(x)]=sec⁡2(x)\frac{d}{dx}[\tan(x)]=\sec^2(x)
  4. ddx[csc⁡(x)]=−csc⁡(x)cot⁡(x)\frac{d}{dx}[\csc(x)]=-\csc(x)\cot(x)
  5. ddx[sec⁡(x)]=sec⁡(x)tan⁡(x)\frac{d}{dx}[\sec(x)]=\sec(x)\tan(x)
  6. ddx[cot⁡(x)]=−csc⁡2(x)\frac{d}{dx}[\cot(x)]=-\csc^2(x)

Inverse Trigonometric Functions

  1. ddx[sin⁡−1(x)]=11−x2\frac{d}{dx}[\sin^{-1}(x)]=\frac{1}{\sqrt{1-x^2}}
  2. ddx[cos⁡−1(x)]=−11−x2\frac{d}{dx}[\cos^{-1}(x)]=-\frac{1}{\sqrt{1-x^2}}
  3. ddx[tan⁡−1(x)]=11+x2\frac{d}{dx}[\tan^{-1}(x)]=\frac{1}{1+x^2}
  4. ddx[csc⁡−1(x)]=−1∣x∣x2−1\frac{d}{dx}[\csc^{-1}(x)]=-\frac{1}{|x|\sqrt{x^2-1}}
  5. ddx[sec⁡−1(x)]=1∣x∣x2−1\frac{d}{dx}[\sec^{-1}(x)]=\frac{1}{|x|\sqrt{x^2-1}}
  6. ddx[cot⁡−1(x)]=−11+x2\frac{d}{dx}[\cot^{-1}(x)]=-\frac{1}{1+x^2}

Exponential and Logarithmic Functions

  1. ddx(ex)=ex\frac{d}{dx}(e^x)=e^x
  2. ddx(ax)=axln⁡(a)\frac{d}{dx}(a^x)=a^x\ln(a)
  3. ddx[ln⁡(∣x∣)]=1x\frac{d}{dx}[\ln(|x|)]=\frac{1}{x}
  4. ddx[log⁡a(x)]=1xln⁡(a)\frac{d}{dx}[\log_a(x)]=\frac{1}{x\ln(a)}

Hyperbolic Functions

  1. ddx[sinh⁡(x)]=cosh⁡(x)\frac{d}{dx}[\sinh(x)]=\cosh(x)
  2. ddx[cosh⁡(x)]=sinh⁡(x)\frac{d}{dx}[\cosh(x)]=\sinh(x)
  3. ddx[tanh⁡(x)]=sech⁡2(x)\frac{d}{dx}[\tanh(x)]=\operatorname{sech}^2(x)
  4. ddx[csch⁡(x)]=−csch⁡(x)coth⁡(x)\frac{d}{dx}[\operatorname{csch}(x)]=-\operatorname{csch}(x)\coth(x)
  5. ddx[sech⁡(x)]=−sech⁡(x)tanh⁡(x)\frac{d}{dx}[\operatorname{sech}(x)]=-\operatorname{sech}(x)\tanh(x)
  6. ddx[coth⁡(x)]=−csch⁡2(x)\frac{d}{dx}[\coth(x)]=-\operatorname{csch}^2(x)

Inverse Hyperbolic Functions

  1. ddx[sinh⁡−1(x)]=1x2+1\frac{d}{dx}[\sinh^{-1}(x)]=\frac{1}{\sqrt{x^2+1}}
  2. ddx[cosh⁡−1(x)]=1x2−1, (x>1)\frac{d}{dx}[\cosh^{-1}(x)]=\frac{1}{\sqrt{x^2-1}}\text{, }(x>1)
  3. ddx[tanh⁡−1(x)]=11−x2, (∣x∣<1)\frac{d}{dx}[\tanh^{-1}(x)]=\frac{1}{1-x^2}\text{, }(|x|<1)
  4. ddx[csch⁡−1(x)]=−1∣x∣1+x2, (x≠0)\frac{d}{dx}[\operatorname{csch}^{-1}(x)]=-\frac{1}{|x|\sqrt{1+x^2}}\text{, }(x\ne0)
  5. ddx[sech⁡−1(x)]=−1x1−x2, (0<x<1)\frac{d}{dx}[\operatorname{sech}^{-1}(x)]=-\frac{1}{x\sqrt{1-x^2}}\text{, }(0<x<1)
  6. ddx[coth⁡−1(x)]=11−x2, (∣x∣>1)\frac{d}{dx}[\coth^{-1}(x)]=\frac{1}{1-x^2}\text{, }(|x|>1)

How to Use This Table

This reference page contains many of the most common differentiation formulas used throughout Calculus. Most rules are written using xx, but the same patterns apply to any variable.

Derivatives describe rates of change and slopes of tangent lines. Many differentiation problems can be solved directly using one of the rules below, while others require combining several rules together.

To see a comprehensive list of antiderivatives, check out our Table of Integrals

Additional Notes