Derivatives

Derivative of a function f at a number a

Notation

We denote the derivative of a function $f$ at a number $a$ as $f^{\prime }(a)$.

Definition

The derivative of a function $f$ at a number $a$ a is given by the following limit (if it exists):
Template:Center top$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$Template:Center bottom
An analogous equation can be defined by letting $x=(a+h)$. Then $h=(x-a)$, which shows that when $x$ approaches $a$, $h$ approaches $0$:
Template:Center top$f^{\prime }(a)=\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a}$Template:Center bottom

Interpretations

As the slope of a tangent line

Given a function $y=f(x)$, the derivative $f^{\prime }(a)$ can be understood as the slope of the tangent line to $f(x)$ at $x=a$:
Template:Center topFile:Derivative as tangent.jpgTemplate:Center bottom

As a rate of change

The derivative of a function $f(x)$ at a number $a$ can be understood as the instantaneous rate of change of $f(x)$ when $x=a$.

At a tangent to one point of a curve

Vocabulary

The point A(a; f(a)) is the point in contact of the tangent and Cf.

Definition

If f is differentiable in a, then the curve C admits at a point A which has for coordinates (a ; f(a)), a tangent : it is the straight line passing by A and of direction coefficient f'(a). An equation of that tangent is written: y = f'(a)*(x-a)+f(a)

Degrees

First Degree Derivative [First Order Derivative; f'(x)]

The first degree derivative of a function, commonly showing the slope of the tangent line at one point of the function, shows its instantaneous rate of change. Intuitively, the first degree derivative reveals the direction of the function; a positive first degree derivative shows the increasing of a function, and it shows decreasing when negative.

Minimum/Maximum

Local

Local minimums/maximums are found from solving for f'(x)=0, for f(x) is differentiable for all x on desired interval. When the first order derivative is zero, it suggests the stop in increasing or decreasing. Whether the point x at f'(x)=0 is a maximum or minimum requires the derivative to the left and right of x. If the derivative is positive on the left and negative on the right; if the graph shifts from increasing to decreasing at point x, f(x) at x would be a local maximum. Conversely, If the derivative is negative on the left and positive on the right; if the graph shifts from decreasing to increasing at point x, f(x) at x would be a local maximum.

Global

Global minimums/maximums are found at the smallest/largest y-values of each graph.

A saddle point refers to the point where f'(x)=0 but the graph maintains its movement direction.

Application

In a function that measures an object's disposition, the first order derivative shows its instantaneous change in position, i.e. velocity.

Second Degree Derivative [Second Order Derivative; f''(x)]

The second order derivative takes

Inflection Point

An inflection point refers to the point where the function changes its concavity (from sloping up to down, vice versa). The inflection point is found from solving for f''(x)=0, for f(x) is differentiable for all x on desired interval.

Application

In a function that measures an object's velocity, the first order derivative shows its instantaneous change in velocity, i.e. acceleration. Template:Sfoot

Differentiation Rules

Power Rule

One of the most commonly used derivative rules for functions in the form of $x^n$($x$ raised to the $n$th power), $n\in ℝ$.

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

Polynomial Differentiations

Taking the derivative of a polynomial requires the differentiation procedure to be applied to each term including the variable $x$.

Example:

$f(x)=x^5+3x^2$

$\frac{d}{dx}f(x)=5x^4+6x$

Negative Power Differentiations

Taking the derivative of a function in the form $\frac{1}{x^n}$ can be achieved through rewriting the function with a negative power, $x^{-n}$.

Example:

$f(x)=\frac{1}{x^7}=x^{-7}$

$\frac{d}{dx}f(x)=-7x^{-8}$

Fractional Exponents and Radicals

Differentiating a function with a radical such as a square root,$\sqrt{x}$, can be done through rewriting the function in the form with a fraction as the exponent, $x^{1/2}$.

Example:

$f(x)=\sqrt{x}=x^{1/2}$

$\frac{d}{dx}f(x)=\frac{1}{2}{x}^{\frac{-1}{2}}=\frac{1}{2\sqrt{2}}$

Quotient Rule

This rule is used to differentiate functions written in the form of $f(x)=\frac{h(x)}{g(x)}$.

$f^{\prime }(x)=\frac{h^{\prime }(x)g(x)-g^{\prime }(x)h(x)}{g(x{)}^2}$

Example:

$f(x)=\frac{x^2+14x^5-\sqrt{2}}{\sqrt{14}+x^{-2}}$

$f^{\prime }(x)=\frac{-2x^{-3}(x^2+14x^5-\sqrt{2})-(2x+70x^4)(\sqrt{14}+x^{-2})}{(\sqrt{14}+x^{-2}{)}^2}$

Trigonometric Functions

Rules for differentiating trigonometric functions:

$\frac{d}{dx}(sin(x))=cos(x)dx$

$\frac{d}{dx}(cos(x))=-sin(x)dx$

$\frac{d}{dx}(tan(x))=se{c}^2(x)dx$

$\frac{d}{dx}(cot(x))=-cose{c}^2(x)dx$

$\frac{d}{dx}(sec(x))=sec(x)tan(x)dx$

$\frac{d}{dx}(csc(x))=-cosec(x)cot(x)dx$

Logarithmic Functions

Rules for differentiating logarithmic functions:

$\frac{d}{dx}ln(x)=\frac{1}{x}dx$

$\frac{d}{dx}{\log }_b(x)=\frac{1}{ln(b)x}dx$

Sample Problems

Differentiate the following:

1. $f(x)=2$
2. $f(x)=x$
3. $f(x)=x^{\frac{1}{2}}+x^5-\sqrt{x}$
4. $f(x)=\frac{-(x+1)}{x^2}$
5. $f(x)=2sin(3x)$
6. $f(x)=\frac{2sin(3x)}{cos(2x)}$
7. $f(x)=sec(3x)tan(2x)$
8. $f(x)=\frac{lnx}{4cosec(x)}$