Derivative of Polynomials and Trigonometric Functions 

1.0Introduction

Differentiation is one of the core concepts of Calculus in JEE Mathematics. The derivative measures the rate of change of a function with respect to a variable.

• For polynomial functions, differentiation gives algebraic expressions useful for slopes, tangents, and optimization problems.
• For trigonometric functions, derivatives are essential in solving problems related to periodicity, maxima and minima, and advanced calculus applications.

In JEE, students must be fluent in differentiating polynomials, trigonometric functions, and their combinations using basic rules.

2.0Basics of Differentiation

If f(x) is a function, then its derivative is defined as:

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

This definition is the foundation, but in practice we use rules of differentiation to calculate derivatives quickly.

Some key rules:

• $\text{Power Rule:}\frac{d}{dx}(x^n)=n{x}^{n-1}$
• $\text{Constant Rule:}\frac{d}{dx}(c)=0$
• $\text{Sum Rule:}\frac{d}{dx}(f+g)=f^{\prime }+g^{\prime }$
• $\text{Product Rule:}\frac{d}{dx}(fg)=f^{\prime }g+f{g}^{\prime }$
• $\text{Quotient Rule:}\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$
• $\text{Chain Rule:}\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

3.0Derivatives of Polynomial Functions

Definition

Polynomial functions are of the form:

$P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$

The derivative is obtained by applying the power rule term by term.

Rules & Formulas

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

$\frac{d}{dx}(a{x}^n)=an{x}^{n-1}$

$\frac{d}{dx}(a_0)=0$

Examples

1. $f(x)=5x^3-2x^2+7x-4$

$f^{\prime }(x)=15x^2-4x+7$

2. $f(x)=x^5$

$f^{\prime }(x)=5x^4$

3. $f(x)=3x^2+4x+1$

f'(x) = 6x + 4

4.0Derivatives of Trigonometric Functions

Standard Results

The basic derivatives to remember:

• $\frac{d}{dx}(\sin x)=\cos x$
• $\frac{d}{dx}(\cos x)=-\sin x$
• $\frac{d}{dx}(\tan x)={\sec }^{2}x,x\ne (2n+1)\frac{\pi }{2}$
• $\frac{d}{dx}(\cot x)=-{\csc }^{2}x,x\ne n\pi$
• $\frac{d}{dx}(\sec x)=\sec x\tan x,x\ne (2n+1)\frac{\pi }{2}$
• $\frac{d}{dx}(\csc x)=-\csc x\cot x,x\ne n\pi$

Derivative Rules

Using product, quotient, and chain rules, we can find derivatives of composite trigonometric functions.

• Example (Product Rule):

$f(x)=x\sin x⟹f^{\prime }(x)=1\cdot \sin x+x\cos x=\sin x+x\cos x$

• Example (Quotient Rule):

$f(x)=\frac{\sin x}{x}⟹f^{\prime }(x)=\frac{x\cos x-\sin x}{x^2}$

• Example (Chain Rule):

$f(x)=\sin (3x)⟹f^{\prime }(x)=3\cos (3x)$

5.0Properties of Derivatives

1. Derivative represents slope of tangent at a point.
2. If f′(x)>0, the function is increasing; if f′(x)<0, it is decreasing.
3. Derivatives help find maxima and minima of functions.
4. For periodic functions like sine and cosine, derivatives remain periodic.
5. Differentiability implies continuity, but the reverse is not always true.

6.0Solved Examples on Derivative of Polynomials and Trigonometric Functions

Example 1: Polynomial Derivative

Find the derivative of $f(x)=2x^4-3x^3+5x-7$

Solution:

$f^{\prime }(x)=8x^3-9x^2+5$

Example 2: Trigonometric Function

Differentiate f(x)=sin⁡2x.

Solution:

$f(x)=sin2x\Rightarrow f\prime (x)=2sinx\cdot cosx=sin(2x)$

Example 3: Mixed Function

Differentiate $f(x)={\sin }^{2}x$

Solution:
Using product rule:

$f^{\prime }(x)=2x\cos x-x^2\sin x$

Example 4: Chain Rule

Differentiate f(x)=cos⁡(5x).

Solution:

$f^{\prime }(x)=-\sin (5x)\cdot 5=-5\sin (5x)$

Example 5: Quotient Rule

Differentiate $f(x)=\frac{1}{\sin x}$

Solution:

$f(x)=\csc x⟹f^{\prime }(x)=-\csc x\cot x$

7.0Practice Questions on Derivative of Polynomials and Trigonometric Functions

1. Differentiate $f(x)=3x^5+2x^3-7x$
2. Find derivative of f(x) = sin⁡(2x) + cos⁡(3x). 
3. Differentiate f(x)=x tan⁡x.
4. Find derivative of $f(x)=\frac{\cos x}{x^2}$
5. If find $f\prime (x).f(x)=x^3\sin x,\text{ find }{f}^{\prime }(x)$