Derivatives

Notation
We denote the derivative of a function $$f$$ at a number $$a$$ as $$f'(a)\,\!$$.

Definition
The derivative of a function $$f$$ at a number $$a$$ a is given by the following limit (if it exists): $$f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$$ An analogous equation can be defined by letting $$x=(a+h)$$. Then $$h=(x-a)$$, which shows that when $$x$$ approaches $$a$$, $$h$$ approaches $$0$$: $$f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$$

As the slope of a tangent line
Given a function $$y=f(x)\,\!$$, the derivative $$f '(a)\,\!$$ can be understood as the slope of the tangent line to $$f(x)$$ at $$x=a$$:

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 $$x=a$$.

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)

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.

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.

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), $$n\in \mathbb{R}$$.

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

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

Example:

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

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

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

Example:

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

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

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:

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

$${d \over dx} f(x)= {1\over2}x^{-1\over2}={1\over {2 \sqrt{2}}}$$

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

$$f'(x) = {{h'(x)g(x)-g'(x)h(x)}\over{g(x)^2}}$$

Example:

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

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

Trigonometric Functions
Rules for differentiating trigonometric functions:

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

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

$${d \over dx}(tan(x))=sec^2(x)dx $$

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

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

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

Logarithmic Functions
Rules for differentiating logarithmic functions:

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

$${d \over dx}\log_b(x)={1 \over {ln(b)x}}dx $$

Sample Problems
Differentiate the following:

$$ $$ $$ $$ $$ $$ $$ $$
 * 1) $$f(x)= 2
 * 1) $$f(x)= x
 * 1) $$f(x)= {x^{1\over2}+x^5-\sqrt{x}}
 * 1) $$f(x)= {{-(x+1)}\over{x^2}}
 * 1) $$f(x)= 2sin(3x)
 * 1) $$f(x)= {{2sin(3x)}\over{cos(2x)}}
 * 1) $$f(x)= sec(3x)tan(2x)
 * 1) $$f(x)= {{lnx}\over4cosec(x)}