Derivative

Introduction

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

--Wikipedia 2023

Basics

A function of a real variable $f(x)$ is differentiable at a point $a$ of its domain, if its domain contains an open interval $I$ containing $a$ and the limit

$$\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}$$

exists. This means that, for every positive real number $ϵ$, there exists a real positive number $\delta$ such that, for every $h$ such that $0<|h|<\delta$, the $f(a+h)$ is defined and

$$|\frac{f(a+h)-f(a)}{h}-L|<ϵ$$

where the vertical bars denote the absolute value.

Rules of Computation

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules of differentiation.

Rules for Basic Functions

Here are the rules for the derivatives of the most common basic functions, where $a$ is a real number.

Derivatives of Powers

$$\frac{d}{dx}{x}^n=n{x}^{n-1}$$

Derivatives of Exponential and Logarithmic Functions

$$\frac{d}{dx}{e}^x=e^x$$ $$\frac{d}{dx}{a}^x=a^{x}ln(a),a>0$$ $$\frac{d}{dx}ln(x)=\frac{1}{x},x>0$$ $$\frac{d}{dx}lo{g}_a(x)=\frac{1}{xln(a)},x>0,a>0$$

Trigonometric Functions

$$\frac{d}{dx}sin(x)=cos(x)$$ $$\frac{d}{dx}cos(x)=-sin(x)$$ $$\frac{d}{dx}tan(x)=se{c}^2(x)=\frac{1}{co{s}^2(x)}=1+ta{n}^2(x)$$

Rules for Combinations of Functions

Constant Rule

If $f$ is constant, then for all $x$,

$$\frac{d}{dx}f(x)=f^{\prime }(x)=0$$

Sum Rule

For all functions $f$ and $g$ and all real numbers $\alpha$ and $\beta$,

$$(\alpha f+\beta g{)}^{\prime }=\alpha f^{\prime }+\beta g^{\prime }$$

Product Rule

For all functions $f$ and $g$,

$$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$$

As a special case, this rule includes the fact $(\alpha f{)}^{\prime }=\alpha f^{\prime }$ whenever $\alpha$ is a constant, because $\alpha f=0\cdot f=0$ by the constant rule.

Quotient Rule

For all functions $f$ and $g$ at all inputs where $g\ne 0$,

$${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$$

Chain Rule

Applies to composite functions, i.e. functions of functions like $h(g(x))$,

$$f=(h\circ g{)}^{\prime }=h^{\prime }(g(x))\cdot g^{\prime }(x)$$

References

Web Links

• Derivative (from Wikipedia, the free encyclopedia)

Note Links

• Calculus
• Limits (Mathematical)