Standard results for derivatives

Algebraic derivative rules

Constant Multiple Rule: \(\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))\)

Sum Rule: \((u + v)' = u' + v'\)

Difference Rule: \((u - v)' = u' - v'\)

Product Rule: \((uv)' = u'v + uv'\)

Quotient Rule:

\[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\]

Chain Rule:

If \(y = f(u)\) and \(u = g(x)\), then

\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]

Alternatively, if \(h(x) = f(g(x))\), then

\[h'(x) = f'(g(x)) \cdot g'(x)\]

Standard derivatives

Constant rule

\[\frac{d}{dx}(c) = 0\]

Power rule

\[\frac{d}{dx}(x^n) = nx^{n-1}\]

Trigonometric functions

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

Inverse trigonometric functions

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

Exponential and logarithmic functions

\[ \begin{align*} & \frac{d}{dx}(e^x) = e^x \\ & \frac{d}{dx}(a^x) = a^x \ln a \\ & \frac{d}{dx}(\ln x) = \frac{1}{x}, x > 0 \\ & \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}, x > 0 \end{align*} \]