Basic Differentiation Rules#
The following rules are useful for computing derivatives of common functions. Let \(f\) and \(g\) be differentiable, \(C\) a constant:
Constant: \(C\) is a constant, \(x\) is a variable.
\[ D(C) = 0 \]Power: \(n\) is a real number.
\[ D(x^n) = n\,x^{n-1} \]Exponential: \(e\) is the base of the natural logarithm.
\[ D(e^x) = e^x \]Logarithm: \(x>0\).
\[ D(\ln x) = \frac{1}{x} \]Trigonometric: \(x\) is a variable.
\[ D(\sin x) = \cos x,\quad D(\cos x) = -\sin x,\quad D(\tan x) = \sec^2 x \]Inverse trigonometric: \(x>0\).
\[ D(\operatorname{arcsin} x) = \frac{1}{\sqrt{1-x^2}},\quad D(\operatorname{arccos} x) = -\frac{1}{\sqrt{1-x^2}},\quad D(\operatorname{arctan} x) = \frac{1}{1+x^2} \]Hyperbolic: \(x\) is a variable.
\[ D(\sinh x) = \cosh x,\quad D(\cosh x) = \sinh x,\quad D(\tanh x) = \operatorname{sech}^2 x \]Inverse hyperbolic: \(x>0\).
\[ D(\operatorname{arcsinh} x) = \frac{1}{\sqrt{1+x^2}},\quad D(\operatorname{arccosh} x) = \frac{1}{\sqrt{x^2-1}},\quad D(\operatorname{arctanh} x) = \frac{1}{1-x^2} \]Absolute value: \(x\) is a variable.
\[\begin{split} D(|x|) = \begin{cases} 1 & \text{if } x>0 \\ -1 & \text{if } x<0 \end{cases} \end{split}\]Step function: \(x\) is a variable.
\[\begin{split} D(\text{sgn}(x)) = \begin{cases} 1 & \text{if } x>0 \\ -1 & \text{if } x<0 \end{cases} \end{split}\]Heaviside step function: \(x\) is a variable.
\[\begin{split} D(H(x)) = \delta(x) = \begin{cases} 0 & \text{if } x\neq0 \\ \infty & \text{if } x=0 \end{cases} \end{split}\]where \(\delta(x)\) is the Dirac delta function.
Dirac delta function: \(x\) is a variable.
\[\begin{split} D(\delta(x)) = \delta'(x) = \begin{cases} 0 & \text{if } x\neq0 \\ \infty & \text{if } x=0 \end{cases} \end{split}\]where \(\delta'(x)\) is the derivative of the Dirac delta function.
Constant multiple
\(\displaystyle D\bigl[C\,f(x)\bigr] = C\,f'(x)\).Sum rule
\(\displaystyle D\bigl[f(x)+g(x)\bigr] = f'(x)+g'(x)\).Product rule
\(\displaystyle D\bigl[f(x)\,g(x)\bigr] = f(x)\,g'(x) + g(x)\,f'(x)\).Quotient rule
\[ D\!\Bigl[\tfrac{f(x)}{g(x)}\Bigr] = \frac{g(x)\,f'(x)\;-\;f(x)\,g'(x)}{[g(x)]^2}. \]