What is the Derivative of xcos x?

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

What is the derivative of xcos x?

Th derivative of xcos x with respect to the variable ‘x’ is equal to -sin x. It is denoted by ddx x . It is the rate of change of the trigonometric function cos s. In a triangle, it is the ratio of adjacent to hypotenuse. It is written as;

$\cos x=\frac{\text{Base}}{\text{Hypotenuse}}$

Derivative of xcos x formula

The formula of derivative xcos x is equal to the negative of the sine function, that is;

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

Where, the derivative of cos x is the major part of the formula.

How do you prove the derivative of xxcos x?

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

1. First Principle
2. Chain Rule
3. Quotient Rule

Derivative of xcos x by first principle

The derivative first principle says that the derivative xcos x is equal to the negative of sin x. 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 xcos x by first principle

To prove the derivative of xcos x by using first principle, replace f(x) by xcos x. 

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

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

Simplifying,

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

Using the trigonometric formula, cos a – cos b = 2sin(a+b)/2 sin(a-b)/2,

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

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

Since we know that sin(-x)=-sin x

$f’(x)=-\lim_{h\to 0}\left(x\sin\frac{2x+h}{2}\frac{\sin\frac{h}{2}}{(h/2)}\right)+ \cos x$

As h approaches zero, sin h/2 h/2 becomes 1. So,

$f’(x)=-x\sin(x+0)+\cos x$

$f’(x)=-x\sin x + \cos x$

Hence the derivative of xcos x is equal to -sin x + cos x.

Derivative of xcos x by chain rule

The derivative of tan x can be calculated by using chain rule because the cosine function can be written as the combination of two functions. The chain rule of derivatives is defined as;

$\frac{dy}{dx}= \frac{dy}{du} × \frac{du}{dx}$

The above rule can be used for differentiation of trig functions.

Proof of derivative of xcos x by chain rule 

To prove the derivative of xcos x by using chain rule, we need to differentiate the function with respect to x. For this, suppose that,

$y=x\cos x$

Differentiate with respect to x and the derivative of x is 1.

$\frac{d}{dx}(x\cos x)=1.\cos x+ x(-\sin x)$

That is,

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

Hence the chain rule also proved that the derivative of xcos x is always cos x–sin x.

Derivative of xcos x using quotient rule

Since the tangent is the ratio of two trigonometric ratios sine and cosine. Therefore, the derivative of tangent can also be calculated by using the quotient rule calculator. The quotient rule is defined as;

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f(x).g'(x)-g(x).f'(x)}{(g(x))^2}$

Proof of derivative of xcos x by quotient rule 

 To prove the derivative of cosine, we can write it as,

$\frac{d}{dx}(x\cos x)=\frac{d}{dx}\left(\frac{x}{\sec x}\right)$

By using quotient rule and since the derivative of sec x is, 

$\frac{d}{dx}(x\cos x)=\frac{1.\sec x – x\sec x\tan x}{\sec^2x}$

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

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

$\frac{d}{dx}(x\cos x)=\frac{1}{\sec x} – \frac{x\tan x}{\sec x}$

Therefore,

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

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

How to find the derivative of xcos x with a calculator?

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

1. Write the function as xcos 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 xcos x.
2. Now, select the variable by which you want to differentiate xcos x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate cosine x. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button. After this step, you will get the derivative of x cosine x within a few seconds.