What is the Derivative of xex?

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

What is the derivative of xe to the x?

The derivative of xe^x, represented by d/dx (xex), is equal to e^x(x+1). This formula is derived using the chain rule of differentiation. It represents the rate of change of the exponential function e^x multiplied by x. Differentiating xe^x is important for determining the slope of the tangent line at any point on the curve. Whether you need to find the second derivative of xe^x or differentiate it using first principles, understanding the basic formula is important.

Derivative of xe^x formula

The formula for differentiating xe^x is the sum of e^x and xe^x, which is represented mathematically as:

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

This formula can be derived using the chain rule of differentiation. It represents the rate of change of the exponential function e^x multiplied by x.

How do you prove the derivative of xex?

There are various methods to derive derivatives of xe^x. Some of the following methods and differentiation rules are;

1. Product Rule
2. Quotient Rule

Each method provides a different way to compute the xex derivative. By using these methods, we can mathematically prove the formula for finding the derivative of xe^x.

Derivative of xe^x by first principle

According to the first principle of derivative, the ln xe^x derivative is equal to e^x+xe^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,

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

You can also use the derivative by definition calculator as it follows the limit definition of derivatives.

Proof of derivative of xe^x by first principle

To prove the differentiation of x e^x by using the first principle, we start by replacing f(x) by xe^x. 

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

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

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

$f'(x)=\lim_{h\to 0}\left(\frac{xe^x(e^h-1)}{h} + \frac{he^x(e^h)}{h}\right)$

$f'(x)=\lim_{h\to 0}\frac{xe^x(e^h-1)}{h}+\lim_{h\to 0}\frac{he^x(e^h)}{h}$

Now separating the limits,

$\lim_{h\to 0}\frac{xe^x(e^h-1)}{h}=xe^x \lim_{h\to 0}\frac{(e^h-1)}{h}$

When h approaches to zero, the first limit becomes:

$\lim_{h\to 0}\frac{xe^x(e^h-1)}{h} = xe^x$

Second limit,

$\lim_{h\to 0}\frac{he^x(e^h)}{h} = e^x \lim_{h\to 0}\frac{he^h}{h}$

Hence we have, 

$f'(x) = xe^x + e^x$

Derivative of xe to the x by product rule

Another method to find the derivative xe^x is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the derivative product rule is:

d/dx(uv) = u(dv/dx) + (du/dx)v 

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x

Proof of derivative of xe by product rule 

To prove the derivative of xe^x by using product rule, we start by assuming that,

$f(x) = xe^x$

By using product rule of differentiation,

$f'(x)=\frac{d}{dx}(x*e^x)$

Since the derivative of e^x is e^x.

$f’(x)= x\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x)$

$f’(x)= xe^x + e^x\cdot 1$

$f’(x)= e^x(x + 1)$

Hence, 

$f’(x) = e^x(x + 1)$

The xe^x derivative can also be calculated by using the derivative product rule calculator.

Derivative of xe^x using quotient rule

Since the cotangent is the reciprocal of the tangent. Therefore, the derivative of xe^x can also be calculated by using the quotient rule. The quotient rule is defined as;

$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right)=\frac{u(x)v'(x)-v(x)u'(x)}{(v(x))^2}$

The quotient rule calculator also follows the above rule and provides the calculations of derivatives in a quick and smart way.

Proof of derivative of xe^x by quotient rule 

 To prove the derivative of xe^x, we can start by writing it as,

$f(x) = \frac{xe^x}{1}$

Supposing that u = xex and v = 1. Now by quotient rule,

$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right)=\frac{u(x)v'(x)-v(x)u'(x)}{(v(x))^2}$

$f'(x)=\frac{(x)(e^x) - (e^x)(1)}{(e^x)^2}$

$f’(x)=\frac{xe^x - e^x}{(e^x)^2}$

$f’(x) =\frac{x-1}{e^x}$

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

How to find the differentiation xe^x with a calculator?

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

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

After completing these steps, you will receive the xe^x differentiation within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.