Derivative formulas and rules reference

Product Rule

\[(fg)^\prime = f^\prime g + g^\prime f\]

Quotient Rule

\[(\frac{f}{g})^\prime = \frac{f^\prime g - g^\prime f}{g^2}\]

Chain Rule

\[\frac{d}{dx}(f(g(x))) = f^\prime (g(x))g^\prime (x)\]

Polynomials

\[\begin{align*} \frac{d}{dx}(c) = 0 \qquad \frac{d}{dx}(x) = 1 \qquad \frac{d}{dx}(cx) = c \\ \\\qquad \frac{d}{dx}(x^n) = nx^{n-1} \qquad \frac{d}{dx}(cx^n)=ncx^{n-1} \end{align*}\]

Trigonometry

\[\begin{align*} \frac{d}{dx}(\sin x) = \cos x && \frac{d}{dx}(\sec x) = \sec x \tan x \\ \frac{d}{dx}(\cos x) = -\sin x && \frac{d}{dx}(\csc x) = -\csc x \cot x \\ \frac{d}{dx}(\tan x) = \sec ^2x && \frac{d}{dx}(\cot x) = -\csc^2 x\end{align*}\]

Inverse Trigonometry

\[\begin{align*} \frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}} && \frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}} \\ \frac{d}{dx}(\cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}} && \frac{d}{dx}(\csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}} \\ \frac{d}{dx}(\tan^{-1}x ) = \frac{1}{1+x^2} && \frac{d}{dx}(\cot^{-1}x) = -\frac{1}{1+x^2}\end{align*}\]

Hyperbolic Trigonometry

\[\begin{align*} \frac{d}{dx}(\sinh x) = \cosh x && \frac{d}{dx}(\operatorname{sech}x) = -\operatorname{sech}x \tanh x \\ \frac{d}{dx}(\cosh x) = \sinh x && \frac{d}{dx}(\operatorname{csch}x) = -\operatorname{csch}x \coth x \\ \frac{d}{dx}(\tanh x) = \operatorname{sech}^2 x && \frac{d}{dx} (\coth x) = -\operatorname{csch}^2x \end{align*}\]

Exponential and Logarithmic

\[\begin{align*} \frac{d}{dx}(e^x) = e^x && \frac{d}{dx}(a^x) = a^x\ln(a) \\ \frac{d}{dx}(\ln x) = \frac{1}{x} && \frac{d}{dx}(\log_a(x)) = \frac{1}{x\ln a}\end{align*}\]

for the derivative of \(\ln (x), x>0\) for \(\ln |x|, x\ne 0\)

\[\begin{align*}\frac{d}{dx}(e^{g(x)}) = g^\prime(x)e^{g(x)} && \frac{d}{dx}(\ln g(x)) = \frac{g^\prime(x)}{g(x)} \end{align*}\]

this is just chain rule, but shows up quite frequently

Implicit

\[\frac{d}{dx}\left[x^{n}y^{m} = c \right]\]

\[\begin{align*}&\Rightarrow\quad nx^{n-1}y^{m} + my^{\prime}y^{m-1}x^{n} = 0\\&\Rightarrow\quad y^{\prime}= \frac{-nx^{n-1}y^{m}}{my^{m-1}x^{n}}\end{align*}\]

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\[\frac{d}{dx}[x^{n}+ y^{m}= c]\]

\[\begin{align*}&\Rightarrow \quad nx^{n-1} + my^{m-1}y^{\prime} = 0\\&\Rightarrow\quad y^{\prime}= \frac{-nx^{n-1}}{my^{m-1}}\end{align*}\]

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\[\frac{d}{dx}\left[e^{y^{m}} = \sin(x^{n})\right]\]

\[\begin{align*} &\Rightarrow \quad my^{m-1}y^{\prime}e^{y^{m}} = nx^{n-1}\cos(x^{n})\\ &\Rightarrow\quad y^{\prime} = \frac{nx^{n-1}\cos(x^{n})}{my^{m-1}e^{y^{m}}}\end{align*}\]

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\[\frac{d}{dx}\left[\sin(y^{m}) = e^{x^{2}}\right]\]

\[\begin{align*} &\Rightarrow \quad my^{m-1}y^{\prime}\cos(y^{m}) = nx^{n-1}e^{x^{n}}\\ &\Rightarrow\quad y^{\prime} = \frac{nx^{n-1}e^{x^{n}}}{my^{m-1}\cos(y^{m})}\end{align*}\]