Home page for accesible maths Math 101 Chapter 3: Differentiation 3.4 Definition of the derivative 3.6 Leibniz’s rules for derivatives

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3.5 Differentiation from first principles

In certain cases we can use this definition to calculate the derivative; this is called differentiation from first principles. A function has a derivative when its graph has a tangent.

$\bullet$ If $f^{\prime}(x)>0$ , then $f$ is increasing.

$\bullet$ If $f^{\prime}(x)<0$ , then $f$ is decreasing.

3.5.1 Example

To find $f^{\prime}(a)$ when $f(x)=x^{2}$ .

Solution. The difference quotient is

3.5.2 Example

The modulus function $f(x)=|x|$ is not differentiable at $0$ .

Solution.