Step By Step Calculus » 10.2 - Rules for Derivatives

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Synopsis
There are several very important properties associated with the derivative and differentiability. Given that ff, gg are differentiable at a\in Aa\in A, and c\in \mathbb{R}c\in \mathbb{R} is a constant, then the following are true:
  • ff and gg are continuous at aa.
  • cf+gcf+g is differentiable, and its derivative is (cf+g)^\prime=cf^\prime+g^\prime(cf+g)^\prime=cf^\prime+g^\prime ( Derivative Linearity rule).
  • fgfg is differentiable, and its derivative is (fg)^\prime=fg^\prime+f^\prime g(fg)^\prime=fg^\prime+f^\prime g (Derivative Product rule).
  • \dfrac{1}{g}\dfrac{1}{g} is differentiable (provided g(a)\neq 0g(a)\neq 0) and \left(\dfrac{1}{g}\right)^\prime=-\dfrac{g^\prime}{g^2}\left(\dfrac{1}{g}\right)^\prime=-\dfrac{g^\prime}{g^2} (Derivative Reciprocal rule).
  • \dfrac{f}{g}\dfrac{f}{g} is differentiable (given that g(a)\neq 0g(a)\neq 0) and \left(\dfrac{f}{g}\right)^\prime=\dfrac{f^\prime g-fg^\prime}{g^2}\left(\dfrac{f}{g}\right)^\prime=\dfrac{f^\prime g-fg^\prime}{g^2} (Derivative Quotient rule).
  • f^pf^p is differentiable if p>0p>0 and f(x)\geq 0f(x)\geq 0 around aa, and \left(f^p\right)^\prime=pf^{p-1}f^\prime\left(f^p\right)^\prime=pf^{p-1}f^\prime (Derivative Power rule).
To find the derivative of more complicated functions, it becomes necessary to introduce more rules. Two of the most important are the following:
  • Derivative Chain rule, which states that \displaystyle \frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)\displaystyle \frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x) or \dfrac{dy}{dx}=\dfrac{dy}{du}\dfrac{du}{dx}\dfrac{dy}{dx}=\dfrac{dy}{du}\dfrac{du}{dx} in Leibniz notation.
  • Derivative Inverse Function rule, which states that \dfrac{d}{dx}f^{-1}(x)=\dfrac{1}{f^\prime(f^{-1}(x))}\dfrac{d}{dx}f^{-1}(x)=\dfrac{1}{f^\prime(f^{-1}(x))}, as long as f^\prime(f^{-1}(x))f^\prime(f^{-1}(x)) exists and f^\prime(f^{-1}(x))\neq 0f^\prime(f^{-1}(x))\neq 0.
These rules may seem to be overly complicated and relatively useless at first glance, but they actually allow the development of some very powerful methods of finding derivatives. The chain rule allows you to differentiate something like \sin(x^2)\sin(x^2) once you know the derivatives of the sine function and the function x^2x^2. The inverse function rule lets us deal with things like logarithmic, inverse hyperbolic, and inverse trigonometric functions. It also allows us to find the derivative of an inverse function at a point without bothering to find a formula for that function.
Using these differentiation rules, it is possible to find derivatives of polynomials and rational functions. For example, for polynomials we have the following formula:
\displaystyle \frac{d}{dx}\sum_{k=0}^n a_kx^k=\sum_{k=1}^n ka_kx^{k-1}.
\displaystyle \frac{d}{dx}\sum_{k=0}^n a_kx^k=\sum_{k=1}^n ka_kx^{k-1}.
Using this polynomial differentiation formula and the quotient rule, we can differentiate rational functions.