## What is the Derivative of xsin x?

Derivatives have a wide range of applications in almost every field of engineering and science. The differentiation of xsin x can be calculated by following the rules of differentiation. Or, we can directly find the derivative xsin x by applying the first principle of differentiation. In this article, you will learn what the derivative of xsin x is and how to calculate the derivative of xsin x by using different approaches.

## What is the derivative of xsin x?

The derivative of xsin x with respect to the variable ‘x’ is equal to the negative of cosine function. It is denoted by d / dx(xsin x). It is the rate of change of the trigonometric function xsin x. In a triangle, it is the ratio of the opposite side to the hypotenuse. It is written as;

$\sin x=\frac{\text{opposite side}}{\text{hypotenuse}}$

## Derivative of xsin x formula

The formula of derivative of xsin x is equal to the negative of the cosine function, that is;

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

## How do you prove the derivative of xsin x?

There are numerous ways to derive derivatives of xsin x. Therefore, we can prove the derivative of x sin x by using;

- First Principle
- Product rule
- Quotient Rule

## Derivative of xsin x by first principle

The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

## Proof of derivative of xsin x by first principle

To prove the derivative of xsin x by using first principle, we need to replace f(x) by xsin x.

$f′(x)=\lim_{h\to 0}\frac{f(x+h)−f(x)}{h}$

Substituting the function f(x) = x sin(x), we get:

$f'(x)=\lim_{h\to 0}\frac{(x + h)\sin(x + h)-x\sin(x)}{h}$

Therefore,

$f'(x)=\lim_{h\to 0}\frac{(x + h)(\sin x\cos h+\cos x\sinh)-x\sin x}{h}$

Simplifying,

$f'(x)=\lim_{h\to 0}\frac{x(\sin(x+h) –\sin x)+h\sin(x+h)}{h}$

By using trigonometric sum and product rule,

$f'(x)=\lim_{h\to 0}\frac{2x\cos\frac{x+h+x}{2}\sin\frac{x+h-x}{2} +h\sin(x+h)}{h}$

$f'(x)=\lim_{h\to 0}\frac{2x\cos\frac{2x+h}{2}\sin\frac{h}{2} +h\sin(x+h)}{h}$

Separating the limits

$f'(x)=\lim_{h\to 0} \frac{2x\cos\frac{2x+h}{2}\sin\frac{h}{2}}{h}+\lim_{h\to 0}\sin(x+h)$

As h approaches to zero,

$f'(x)=2x\cos\left(\frac{2x+0}{2}\right)\times\left(\frac{1}{2}\right)+\sin(x+0)$

Hence the derivative of xsin x by using derivative first principle calculator,

$f’(x)=x\cos x+\sin x$

## Derivative of xsin x using quotient rule

Since the function sine is the ratio of opposite to the hypotenuse of a triangle. Therefore, the derivative of sin can also be calculated by using the quotient rule. The quotient rule is defined as;

$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right)=\frac{u(x).v’(x) –v(x).u’(x)}{(v(x))^2}$

## Proof of derivative of xsin x by quotient rule

To prove the differentiation of xsin x, we can write it,

$f(x)=x\sin x= \frac{x}{\csc x}=\frac{u}{v}$

Supposing that u = x and v = cosec x. Now by quotient rule,

$f’(x)=\frac{v(x).u’(x) –u(x).v’(x)}{(v(x))^2}$

$f'(x)=\frac{\csc x\frac{d}{dx}(x) – x\frac{d}{dx}(\csc x)}{(\csc x)^2}$

$f'(x)=\frac{\csc x(1) – x(-\csc x\cot x)}{\csc^2x}$

$f'(x)=\frac{\csc x+x\csc x\cot x}{\csc^2x}$

$f'(x)=\frac{1+x\cot x}{\csc x}$

$f'(x)=\frac{1}{\csc x}+\frac{x\cot x}{\csc x}$

$f'(x)=\sin x + x\cos x$

Hence, we have derived the derivative of xsin x using the quotient rule of differentiation calculator.

## How to find the derivative of xsin x with a calculator?

The easiest way to calculate the derivative of xsin x is by using an online tool. You can use our differentiation calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

- Write the function as xsin x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of xsin x.
- Now, select the variable by which you want to differentiate xsin x. Here you have to choose ‘x’.
- Select how many times you want to differentiate xsin x. In this step, you can choose 2 for second and 3 to find the third derivative.
- Click on the calculate button. After this step, you will get the derivative of xsin x within a few seconds.