Differentiation

Definition

limit of the difference quotient: $f^{\prime }(x_0)={\lim }_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}$ OR $f^{\prime }(x_0)={\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}\left(=\frac{\Delta f(x)}{\Delta x}\right)$

Notation

When $y=f(x)$, $\frac{d}{dx}f(x)=\frac{dy}{dx}=\frac{df}{dx}=f^{\prime }(x)$

Evaluating Derivatives

• Standard Derivatives
• Rules of Differentiation
• Implicit Differentiation

Differentiability

Definition

$f$ is differentiable at $x=x_0$ When $f^{\prime }(x_0)$ exists If $f$ is differentiable at $x=x_0$, then $f$ is continuous at $x=x_0$ (Continuity of Functions) If $f$ is continuous at $x=x_0$, it may not be differentiable at $x=x_0$

differentiable $\iff$ smooth $\iff$ unique tangent exists

On an interval

$f$ is said to be differentiable on an interval $I$ if it is differentiable at every point in $I$