Formulas and Rules for Derivatives in Differential Calculus
Formulas
\( f(x) \) | \( \dfrac{d f(x)}{dx} \) |
|---|---|
| \( x^n \) | \( n x^{n-1} \) |
| \( e^x \) | \( e^x \) |
| \( b^x \) | \( \ln b \cdot b^x \) |
| \( \ln x \) | \( \dfrac{1}{x} \) |
| \( \log_b x \) | \( \dfrac{1}{ x \ln b} \) |
| \( \sin x \) | \( \cos x \) |
| \( \cos x \) | \( - \sin x \) |
| \( \tan x \) | \( \sec^2 x \) |
| \( \cot x \) | \( - \csc^2 x \) |
| \( \sec x \) | \( \sec x \tan x \) |
| \( \csc x \) | \( - \csc x \cot x\) |
| \( \sin^{-1} x\) | \( \dfrac{1}{\sqrt{1-x^2}} \) |
| \( \cos^{-1} x\) | \( - \dfrac{1}{\sqrt{1-x^2}} \) |
| \( \tan^{-1} x\) | \( \dfrac{1}{1+x^2} \) |
| \( \sinh x \) | \( \cosh x \) |
| \( \cosh x \) | \( \sinh x \) |
| \( \tanh x \) | \( \text{sech}^2 x \) |
| \( \coth x \) | \( - \text{csch}^2 x \) |
| \( \text{sech} \; x \) | \( -\text{sech} \; x \tanh x \) |
| \( \text{csch} \; x \) | \( - \text{csch} \; x \coth x\) |
| \( \sinh^{-1} x\) | \( \dfrac{1}{\sqrt{x^2+1}} \) |
| \( \cosh^{-1} x\) | \( \dfrac{1}{\sqrt{x^2-1}} \) |
| \( \tanh^{-1} x\) | \( \dfrac{1}{1-x^2} \) |
| \( \coth^{-1} x\) | \( \dfrac{1}{1-x^2} \) |
Rules
Let \( u \) and \( v \) be two functions with derivatives \( u' \) and \( v' \)
- Linearity: \( \dfrac{d(a u + b v)}{dx} = a u' + b v' \) , \( a \) and \( b \) are constants
- Product Rule: \( \dfrac{d(u v)}{dx} = u' v + u v' \)
- Quotient Rule: \( \dfrac{d\left(\dfrac{u}{v}\right)}{dx} = \dfrac{u' v - u v'}{v^2} \)
- Chain Rule: \( \dfrac{d(u (v) )}{dx} = \dfrac{du}{dv} \dfrac{dv}{dx} \)
- Power Rule for a Function: \( \dfrac{d(u^n)}{dx} = n u'u^{n-1} \)
- Derivative of a Function Raised to the Power of Another Function: \( \dfrac{d(u^v)}{dx} = \left(v' \ln (u) + \dfrac{u'}{u} v \right) u^v \)
More References and Links
Handbook of Mathematical Functions
Mathematics for Engineers with Examples and Solutions