Derivative Formulas

General Rules

${\frac {\mathrm {d} }{\mathrm {d} x}}(f+g)={\frac {\mathrm {d} f}{\mathrm {d} x}}+{\frac {\mathrm {d} g}{\mathrm {d} x}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}(c\cdot f)=c\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}(f\cdot g)=f\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}+g\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {f}{g}}\right)={\dfrac {-f\cdot {\dfrac {\mathrm {d} g}{dx}}+g\cdot {\dfrac {\mathrm {d} f}{\mathrm {d} x}}}{g^{2}}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}[f(g(x))]={\frac {\mathrm {d} f}{\mathrm {d} g}}\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}=f'(g(x))\cdot g'(x)$

${\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}f(x)g(x)=\sum _{i=0}^{n}\left({\begin{matrix}n\\i\end{matrix}}\right)f^{(n-i)}(x)g^{(i)}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {1}{f}}\right)=-{\frac {f'}{f^{2}}}$

Powers and Polynomials

• ${\frac {\mathrm {d} }{\mathrm {d} x}}(c)=0$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}x=1$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}x^{n}=nx^{n-1}$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}$
• ${{\frac {\mathrm {d} }{\mathrm {d} x}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}}$

Trigonometric Functions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\sin(x)=\cos(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\cos(x)=-\sin(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\tan(x)=\sec^2(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\cot(x)=-\csc^2(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\sec(x)=\sec(x)\tan(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\csc(x)=-\csc(x)\cot(x)}

Exponential and Logarithmic Functions

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}a^x=a^x\ln(a)\qquad\text{if }a>0}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\ln(x)=\frac{1}{x}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\log_a(x)=\frac{1}{x\ln(a)}\qquad\text{if }a>0\ ,\ a\ne1}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}(f^g)=\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{g\ln(f)}\right)=f^g\left(f'\frac{g}{f}+g'\ln(f)\right)\ ,\qquad f>0}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}(c^f)=\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{f\ln(c)}\right)=c^f\ln(c)\cdot f'}

Inverse Trigonometric Functions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\arcsin(x)=\frac{1}{\sqrt{1-x^2}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\arccos(x)=-\frac{1}{\sqrt{1-x^2}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\arctan(x)=\frac{1}{x^2+1}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\arccot(x)=-\frac{1}{x^2+1}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\arcsec(x)=\frac{1}{|x|\sqrt{x^2-1}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\arccsc(x)=-\frac{1}{|x|\sqrt{x^2-1}}}

Hyperbolic and Inverse Hyperbolic Functions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\sinh(x)=\cosh(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\cosh(x)=\sinh(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\tanh(x)={\rm sech}^2(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm sech}(x)=-\tanh(x){\rm sech}(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\coth(x)=-{\rm csch}^2(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm csch}(x)=-\coth(x){\rm csch}(x)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm arsinh}(x)=\frac{1}{\sqrt{1+x^2}}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm arcosh}(x)=\frac{1}{\sqrt{x^2-1}}\ ,\ x>1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm artanh}(x)=\frac{1}{1-x^2}\ ,\ |x|<1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm arcsch}(x)=-\frac{1}{|x|\sqrt{1+x^2}}\ ,\ x\ne0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm arsech}(x)=-\frac{1}{x\sqrt{1-x^2}}\ ,\ 0<x<1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{\rm arcoth}(x)=\frac{1}{1-x^2}\ ,\ |x|>1}

Resources

• Table of Derivatives, Wikibooks: Calculus
• Derivative Function & Interpretations. PowerPoint file created by Professor Cynthia Roberts, UTSA.
• Limits & Derivative Formulas. PowerPoint file created by Professor Cynthia Roberts, UTSA.
• Exponential and Logarithms. PowerPoint file created by Professor Cynthia Roberts, UTSA.