derivatives

A derivative is a ratio of change in the value of the function to change in the independent variable.  
Derivatives of a function at some point denote the rate of change of a function at that point. The rate of change can be calculated by the rate of change of the function \(\Delta y\) to the change of the independent variable \(\Delta x\), this ratio is considered in limit as  \(\Delta x \to 0\). the derivative of a function f(x) represents its rate of change and is denoted by either \(f\prime(x) \) or df ∕ dx

Let’s first look at its definition and a pictorial illustration of the derivative.

Derivative of f is the rate of change of f. Look at the graph of a curve above. It represents the value of f(x) at two points x and \(x + \Delta x \), as f(x) and \(f(x + \Delta x)\) respectively. As you make the interval between these two points smaller until it is infinitesimally small, we have a limit \(\Delta x \to 0\). 

\(f\prime(x) = \frac{df }{dx} = \lim\limits_{ \Delta x\to0}\frac{f(x+\Delta x) - f(x)}{\Delta x} = \lim\limits_{ \Delta x\to0}\frac{\Delta f }{\Delta x}\)

The numerator \(f(x + \Delta x) - f(x)\) represents the corresponding change in the value of the function f over the interval  \(\Delta x\). This makes the derivative of a function f at a point x, the rate of change of f at that point.

Steps to find the derivative of the function f(x) at point x is:

1. Form the difference quotient \(\frac{dy }{dx} = \frac{f(x+\Delta x) - f(x)}{\Delta x}\)
2. Simplify the quotient, canceling where ever possible.
3. Find the derivative \(f\prime(x)\), applying the limit to the quotient. If this limit exists, then we say that function f(x) is differentiable at x.

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Let us try to derive the derivatives for a few functions 

Example 1: Calculate the derivative of the function y = x

\(y\prime(x) = \lim\limits_{ \Delta x\to0}\frac{\Delta y}{\Delta x} = \lim\limits_{\Delta x\to0} \frac{(x+\Delta x) - x}{\Delta x} = \lim\limits_{ \Delta x\to0}\frac{\Delta x}{\Delta x} = \lim\limits_{ \Delta x\to0}1=1\)

Example 2: Find the derivative of the function f(x) = 5x + 2

\(\Delta y = y(x+\Delta x) - y(x) = [5(x+\Delta x) +2] - [5x +2] = 5x + 5\Delta x + 2 - 5x - 2 = 5\Delta x\)

The difference ratio is \(\frac{5\Delta x}{\Delta x}\) = 5

\(\therefore\textrm{ Derivative } f\prime(x) = \lim\limits_{ \Delta x\to0}5 = 5\)

Example 3: Find the derivative of quadratic equation f(x) = x². Let's use the graph and understand derivatives in a better way.

f(x) = x²

\(\Delta y = y(x+\Delta x) - y(x) = (x+\Delta x)^2 - x^2\)

\(\Delta y = x^2 + 2x\Delta x +\Delta x^2-x^2 = \Delta x(2x+\Delta x)\) 

The derivative of x² is 2x. It means that for function x², the rate of change at any point is 2x.

 

the rate of change of f at x = 2 is the value of \(f\prime(x)\) at x = 2, i.e. \(f\prime (x) = 4\) 

Derivatives of common functions

Common function	function	Derivative
Constant	c	0
Line	x	1
	ax	a
Square	x2	2x
Square Root	\(\sqrt x\)	\(\frac{1}{2}x^{-1/2}\)
Exponential	ex	ex
Logarithms	\(\log_a(x)\)	1/(x In(a))
Trigonometry(x in radians)	\(\sin(x)\)	\(\cos(x) \)
	\(\cos(x)\)	\(-\sin(x)\)
	\(\tan(x)\)	\(\sec^2(x)\)

Here are useful rules to help you work out the derivatives of many functions:

• Constant Rule: f(x) = c then \(f\prime(x) = 0\)
• Constant Multiple Rule: \(g(x) = c \cdot f(x) \textrm{ then } g\prime(x) = c \cdot f\prime(x)\)
• Power Rule:\( f(x) = x^n \textrm{ then } f\prime(x) = nx^{n-1}\)
• Sum and Difference Rule: \(h(x) = f(x) \pm g(x) \textrm{ then } h\prime(x) = f\prime(x) \pm g\prime(x)\)
• Product Rule: \(h(x) = f(x)g(x) \textrm{ then } h\prime(x) = f\prime(x)g(x) + f(x)g\prime(x)\)
• Quotient Rule: \(h(x) = \frac{f(x)}{g(x)} \textrm{ then } h\prime(x) = \frac{ f\prime(x)g(x) + f(x)g\prime(x)}{g(x)^2}\)
• Chain Rule: \(h(x) = f(g(x)) \textrm{ then } h\prime(x) = f\prime(g(x))g\prime(x)\)

Note: The slope of a tangent line at a point is its derivative at that point. If a tangent line is drawn for a curve y = f(x) at a point (x₀, y₀), then its slope (m) is obtained by simply substituting the point in the derivative of the function.

Example 4: Differentiate 10x⁵

\(y\prime = \frac{dy}{dx} = \frac{d(10x^5)}{dx}\)

\(10 \times 5 x^4 = 50 x^4\)(applying power rule)

Example 5: Differentiate tan²x

\(y\prime = \frac{dy}{dx} = \frac{d(tan^2x)}{dx}\)

\(2\tan x^{2-1} \times \frac{d(\tan x)}{dx}\)(applying chain rule)

\(2\tan x⋅ \sec ^2 x\)