Derivative

Definition

Derivative

Suppose $f:(a,b)\to \mathbb{R}$, and $c\in (a,b)$ $f^{\prime }(c)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}=\underset{x\to c}{\lim }\frac{f(x)-f(c)}{x-c}=\underset{\Delta \to 0}{\lim }\frac{f(c+\Delta x)-f(c)}{\Delta x}$

If the $f^{\prime }(c)$ exists, then the function $f$ is differentiable at a point $c$

Right-Derivative

Suppose $f:[a,b)\to \mathbb{R}$ $f_{+}^{\prime }(a)=\underset{\Delta \to 0^{+}}{\lim }\frac{f(a+\Delta x)-f(a)}{\Delta x}$

If the $f_{+}^{\prime }(a)$ exists, then the function $f$ is right differentiable at a point $a$

Left-Derivative

Suppose $f:(a,b]\to \mathbb{R}$ $f_{-}^{\prime }(b)=\underset{\Delta \to 0^{-}}{\lim }\frac{f(b+\Delta x)-f(b)}{\Delta x}$

If the $f_{-}^{\prime }(b)$ exists, then the function $f$ is left differentiable at a point $b$

Derivative Function

Suppose $f:\mathcal{D}\to \mathbb{R}$, and $f$ is differentiable $\forall x\in \mathcal{D}$, $f^{\prime }(x)=\frac{df}{dx}=\underset{y\to x}{\lim }\frac{f(y)-f(x)}{y-x}$

$f^{\prime }(x)$ is called the derivative function or the derivative of $f$

Properties

Rules of Computation

If $f,g:\mathcal{D}\to \mathbb{R}$ is differentiable at $a\in \mathcal{D}$, then $f+g,f-g,\frac{f}{g}$ also differentiable, and the following are satisfied

• Linearity: $(af\pm bg{)}^{\prime }(x)=a{f}^{\prime }(x)\pm b{g}^{\prime }(x)$
• product rule: $(fg{)}^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$
• quotient rule: ${\left(\frac{f}{g}\right)}^{\prime }(x)=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{\{g(x){\}}^2}$, where $g(x)\ne 0$
• Chain Rule: $(g\circ f{)}^{\prime }(a)=g^{\prime }(f(a))f^{\prime }(a)$

Facts

If $f$ is differentiable at $a$, then $f$ must also be continuous at $a$

Suppose $f:\mathcal{D}\to \mathbb{R}$ If $f$ has an Extremum at $a\in \mathcal{D}$ and differentiable, then $f^{\prime }(a)=0$