Differentiation Rules

Theorem: Arithmetic Differentiation Rules

If $f,g:D\subseteq \mathbb{R}\to \mathbb{R}$ are differentiable at $x\in D$, then the following hold:

• Linearity: $[\alpha f(x)+\beta g(x){]}^{\prime }=\alpha f^{\prime }(x)+\beta g^{\prime }(x)$ for all $\alpha ,\beta \in \mathbb{R}$

• Product Rule: $[f(x)g(x){]}^{\prime }=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

• Quotient Rule: ${\left[\frac{f(x)}{g(x)}\right]}^{\prime }=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{g(x{)}^2}$ provided that $g(x)\ne 0$

• General Power Rule: ${\left[f(x{)}^{g(x)}\right]}^{\prime }=f(x{)}^{g(x)}\left[f^{\prime }(x)\frac{g(x)}{f(x)}+g^{\prime }(x)\ln (f(x))\right]$

PROOF

Proof of linearity:

$$\begin{array}{rl}[\lambda f(x)+\mu g(x){]}^{\prime }& =\underset{\Delta x\to 0}{\lim }\frac{\lambda f(x_0+\Delta x)+\mu g(x_0+\Delta x)-\lambda f(x_0)-\mu g(x_0)}{\Delta x}\\ & =\underset{\Delta x\to 0}{\lim }\frac{\lambda [f(x_0+\Delta x)-f(x_0)]+\mu [g(x_0+\Delta x)-g(x_0)]}{\Delta x}\\ & =\underset{\Delta x\to 0}{\lim }\lambda \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}+\underset{\Delta x\to 0}{\lim }\mu \frac{g(x_0+\Delta x)-g(x_0)}{\Delta x}\\ & =\lambda \underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}+\mu \underset{\Delta x\to 0}{\lim }\frac{g(x_0+\Delta x)-g(x_0)}{\Delta x}\\ & =\lambda f^{\prime }(x_0)+\mu g^{\prime }(x_0)\end{array}$$

Theorem: Chain Rule

If $g:D\subseteq \mathbb{R}\to \mathbb{R}$ is differentiable at $x\in D$ and $f:g(D)\to \mathbb{R}$ is differentiable at $g(x)$, then

$$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))\cdot g^{\prime }(x)$$

PROOF

TODO

Theorem: Derivative of the Inverse Function

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be invertible.

If $f$ is differentiable at $f^{-1}(y),y\in f(D)$ and $f^{\prime }(f^{-1}(y))\ne 0$, then

$$[f^{-1}(y){]}^{\prime }=\frac{1}{f^{\prime }(f^{-1}(y))}$$

PROOF

TODO