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;

1. First Principle
2. Product rule
3. 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.

1. 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.
2. Now, select the variable by which you want to differentiate xsin x. Here you have to choose ‘x’.
3. 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. 
4. Click on the calculate button. After this step, you will get the derivative of xsin x within a few seconds.