Derivative

DERIVATIVES

A derivative is a concept used to measure the rate of change of a function concerning one of its independent variables. Visually, it signifies the steepness of the tangent line to the function’s graph at a specific point.

If f(x) represents a function of, then the derivative of $f$ concerning $x$ , denoted as f'(x), $\frac{df}{dx}$

DERIVATIVE FORMULA

We have derivative formulas for algebraic, trigonometric, logarithmic, exponential functions and these formulas are derived from the first principle of differentiation.

POWER RULE FOR DERIVATIVE

f'(x) = nx^{n-1}

If $f(x)=x^{n}$ , where $n$ is a constant, then the derivative $f'(x) = nx^{n-1}$

DERIVATIVES OF TRIG FUNCTIONS

Derivative of sin

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

Derivative of cos

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

Derivative of tan

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

Derivative of cot

\frac{d}{dx}(\cot x) = -\csc^2 x

Derivative of sec

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

Derivative of cosec

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

DERIVATIVES OF LOGARITHMIC FUNCTIONS

Derivative of lnx

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

Derivative of logx

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

DERIVATIVES OF EXPONENTIAL FUNCTIONS

Derivative of e

\frac{d}{dx}(e^x) = e^x

DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS

Derivative of arcsin

\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 – x^2}}

Derivative of arccos

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

Derivative of arctan

\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}

Derivative of arccsc

$\frac{d}{dx}(\text{arccsc} x) = -\frac{1}{|x|\sqrt{x^2 – 1}}$

Derivative of arcsec

$\frac{d}{dx}(\text{arcsec} x) = \frac{1}{|x|\sqrt{x^2 – 1}}$

Derivative of arccot

$\frac{d}{dx}(\text{arccot} x) = -\frac{1}{1 + x^2}$

BASIC DERIVATIVE RULES

CONSTANT RULE DERIVATIVE

f'(x) = 0

SUM RULE DERIVATIVE

(f + g)'(x) = f'(x) + g'(x)

DIFFERENCE RULE DERIVATIVE

(f - g)'(x) = f'(x) - g'(x)

PRODUCT RULE DERIVATIVE

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

QUOTIENT RULE DERIVATIVE

\left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) – f(x)g'(x)}{(g(x))^2}

CHAIN RULE DERIVATIVE

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

IMPLICIT DIFFERENTIATION

Implicit differentiation is a method utilized in calculus to determine the derivative of an implicitly defined function. When a function is defined implicitly, it means that the relationship between the variables is not explicitly stated. Instead, the function is expressed through an equation involving both variables.

The primary steps involved in implicit differentiation are as follows:

1. Differentiate Both Sides

2. Apply Chain Rule

3. Isolate the Derivative

4. Simplify if Necessary

For example, consider the equation of a circle: $x^{2}+y^{2}=r^{2}$ denotes the radius of the circle.

Differentiate Both Sides: Differentiate both sides of the equation with respect to $x$

\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(r^2)

Apply Chain Rule: Since $y$ is a function of $x$ , the chain rule is applied when differentiating $y^{2}$

2x + 2y \frac{dy}{dx} = 0

Isolate the Derivative: Solve for $\frac{dy}{dx}$
$\frac{dy}{dx} = -\frac{x}{y}$

PARTIAL DERIVATIVE

Partial derivatives are a fundamental concept in multivariable calculus, enabling us to understand how functions change concerning each of their variables while holding others constant.

In a function $f (x, y)$ of two variables $x$ and ,the partial derivative of $f$ with respect to $x$ , denoted $\frac{\partial f}{\partial x}$ represents how $f$ changes with changes in $x$ , while $y$ remains fixed. Similarly, the partial derivative of $f$ with respect to $y$ , denoted $\frac{\partial f}{\partial y}$ indicates the rate of change of $f$ with changes in $y$ , while $x$ is held constant.

To compute a partial derivative, we differentiate the function with respect to the variable of interest while treating other variables as constants. For instance, for the function $f(x, y) = x^{2} + y^{2}$

The partial derivative $\frac{\partial f}{\partial x}$ is obtained by treating $y$ as constant and differentiating $f$ with respect to $x$ , resulting in $2 x$ .
Similarly, $\frac{\partial f}{\partial y}$ is found by treating $x$ as constant and differentiating $f$ with respect to $y$ , yielding $2 y$

Partial derivatives find wide applications in various fields, including physics, engineering, economics, and optimization problems, where functions depend on multiple variables simultaneously.

PARAMETRIC DERIVATIVES

Parametric derivatives refer to the derivatives of parametric equations, which describe the coordinates of a point in terms of one or more independent variables, often denoted as t.

Given parametric equations x = x(t) and y = y(t), the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ represent the rates of change of x and y with respect to t , respectively.

For example, for the parametric equations $x(t) = \cos(t)$ and $y(t) = \sin(t)$ , we find:

$\frac{dx}{dt} = \sin(t)$

$\frac{dy}{dt} = \cos(t)$

These derivatives describe how the coordinates x and y change as the parameter t varies, providing valuable insights into the motion or trajectory described by the parametric equations.