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Differentiation Rules

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Bycalculus_cl5bvo
General formulas

\frac{\mathrm{d} (c)}{\mathrm{d} x}=0

\frac{\mathrm{d} }{\mathrm{d} x}[cf(x)]=cf'(x)

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

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

\frac{\mathrm{d} }{\mathrm{d} x}[f(x)g(x)]=f(x)g'(x)+g(x)f'(x) (Product Rule)

\frac{\mathrm{d} }{\mathrm{d} x}[\frac{f(x)}{g(x)}]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^{2}} (Quotient Rule)

\frac{\mathrm{d} }{\mathrm{d} x}f(g(x)) = f'(g(x))g'(x) (Chain Rule)

\frac{\mathrm{d} }{\mathrm{d} x}(x^{n})=nx^{n-1} (Power Rule)

Exponential and Logarithmic Functions

\frac{\mathrm{d} }{\mathrm{d} x}(e^{x}) = e^{x}

\frac{\mathrm{d} }{\mathrm{d} x}(a^{x}) = a^{x}ln(x)

\frac{\mathrm{d} }{\mathrm{d} x}ln|x| = \frac{1}{x}

\frac{\mathrm{d} }{\mathrm{d} x}(log_{a}x) = \frac{1}{x*ln(a)}

Trigonometric Functions

\frac{\mathrm{d} }{\mathrm{d} x}(sin(x)) = cos(x)

\frac{\mathrm{d} }{\mathrm{d} x}(cos(x)) = -sin(x)

\frac{\mathrm{d} }{\mathrm{d} x}(tan(x)) = sec^{2}(x)

\frac{\mathrm{d} }{\mathrm{d} x}(csc(x)) = -csc(x)cot(x)

\frac{\mathrm{d} }{\mathrm{d} x}(sec(x)) = sec(x)tan(x)

\frac{\mathrm{d} }{\mathrm{d} x}(cot(x)) = -csc^{2}(x)

Inverse Trigonometric Functions

\frac{\mathrm{d} }{\mathrm{d} x}(sin^{-1}(x)) = \frac{1}{\sqrt{1-x^{2}}}

\frac{\mathrm{d} }{\mathrm{d} x}(cos^{-1}(x)) = -\frac{1}{\sqrt{1-x^{2}}}

\frac{\mathrm{d} }{\mathrm{d} x}(tan^{-1}(x)) = \frac{1}{1+x^{2}}

\frac{\mathrm{d} }{\mathrm{d} x}(csc^{-1}(x)) = -\tfrac{1}{x\sqrt{x^{2}-1}}

\frac{\mathrm{d} }{\mathrm{d} x}(sec^{-1}(x)) = \frac{1}{x\sqrt{x^{2}-1}}

\frac{\mathrm{d} }{\mathrm{d} x}(cot^{-1}(x)) = -\frac{1}{1+x^{2}}

Hyperbolic Functions

\frac{\mathrm{d} }{\mathrm{d} x}(sinh(x)) = cosh(x)

\frac{\mathrm{d} }{\mathrm{d} x}(cosh(x)) = sinh(x)

\frac{\mathrm{d} }{\mathrm{d} x}(tanh(x)) = sech^{2}(x)

\frac{\mathrm{d} }{\mathrm{d} x}(csch(x)) = -csch(x)coth(x)

\frac{\mathrm{d} }{\mathrm{d} x}(sech(x)) = -sech(x)tanh(x)

\frac{\mathrm{d} }{\mathrm{d} x}(coth(x)) = - csch^{2}(x)

Inverse Hyperbolic Functions

\frac{\mathrm{d} }{\mathrm{d} x}(sinh^{-1}(x)) = \frac{1}{\sqrt{1+x^{2}}}

\frac{\mathrm{d} }{\mathrm{d} x}(cosh^{-1}(x)) = \frac{1}{\sqrt{x^{2}-1}}

\frac{\mathrm{d} }{\mathrm{d} x}(tanh^{-1}(x)) = \frac{1}{1-x^{2}}

\frac{\mathrm{d} }{\mathrm{d} x}(csch^{-1}(x)) = -\frac{1}{|x|\sqrt{x^{2}+1}}

\frac{\mathrm{d} }{\mathrm{d} x}(sech^{-1}(x)) = -\frac{1}{x\sqrt{1-x^{2}}}

\frac{\mathrm{d} }{\mathrm{d} x}(coth^{-1}(x)) = \frac{1}{1-x^{2}}

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