Print preview
Handout 36.1 Derivative Rules
Rules for specific types of functions.
- Constant functions
- \(\displaystyle (k)' = 0\)
- Power functions
- \(\displaystyle (x^{n})' = nx^{n-1}\)
- Exponential functions
-
\((a^{x})' = \ln(a) a^{x} \text{ (for } a>0\text{)}\)\((e^{x})' = e^{x}\)
- Logarithmic functions
- \(\displaystyle (\ln(x))' = \frac{1}{x}\)
- Trigonometric functions
-
\((\sin(x))' = \cos(x)\text{.}\)\((\cos(x))' = -\sin(x)\text{.}\)\((\tan(x))' = \sec^{2}(x) = \frac{1}{\cos^{2}(x)}\text{.}\)\((\arcsin(x))' = \frac{1}{\sqrt{1-x^{2}}}\text{.}\)\((\arctan(x))' = \frac{1}{1+x^{2}}\text{.}\)
Rules for combinations of functions.
- Constant multiples
- \(\displaystyle (k\cdot f(x))' = k\cdot f'(x)\)
- Sum and difference
- \(\displaystyle (f(x) \pm g(x))' = f'(x) \pm g'(x)\)
- Products (the product rule)
- \(\displaystyle (f(x)\cdot g(x))' = f'(x)g(x) + f(x)g'(x)\)
- Quotients (the quotient rule)
- \(\displaystyle \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^{2}}\)
- Compositions (the chain rule)
- \(\displaystyle (f(g(x)))' = f'(g(x))\cdot g'(x)\)
