Question

How do you find the derivative of \[y=2x{{e}^{x}}-2{{e}^{x}}\]?

Answer 1

Hint: To solve the given question, we must know the derivatives of some functions and a rule of derivative. The function whose derivative we should know are \[x\And {{e}^{x}}\], their derivative with respect to x are \[1\And {{e}^{x}}\] respectively. We should also know the product rule of the differentiation which states that \[\dfrac{d\left( f(x)g(x) \right)}{dx}=\dfrac{d\left( f(x) \right)}{dx}g(x)+f(x)\dfrac{d\left( g(x) \right)}{dx}\]. We will use these to find the derivative of the given expression.

Complete step-by-step answer:
We are given the expression \[y=2x{{e}^{x}}-2{{e}^{x}}\], we need to find its derivative. The given expression has two terms, as we can see both of the terms have \[2{{e}^{x}}\] common to them. So, we can take this factor common to both of them, and write the given expression as,
\[y=2{{e}^{x}}\left( x-1 \right)\]
The given expression is of the form \[f(x)g(x)\], that is a product of two functions. We know the product rule states that \[\dfrac{d\left( f(x)g(x) \right)}{dx}=\dfrac{d\left( f(x) \right)}{dx}g(x)+f(x)\dfrac{d\left( g(x) \right)}{dx}\]. Here we have \[f(x)=2{{e}^{x}}\And g(x)=x-1\]
To find the \[\dfrac{dy}{dx}\] or \[\dfrac{d\left( 2{{e}^{x}}\left( x-1 \right) \right)}{dx}\], we need to find \[\dfrac{d\left( 2{{e}^{x}} \right)}{dx}\], and \[\dfrac{d\left( x \right)}{dx}\].
We know the derivative of \[{{e}^{x}}\] with respect to x, is \[{{e}^{x}}\] itself. So, \[\dfrac{d\left( 2{{e}^{x}} \right)}{dx}=2{{e}^{x}}\]. Also, the derivative of x with respect to x is 1, \[\dfrac{d\left( x-1 \right)}{dx}=1\].
\[\dfrac{dy}{dx}=\dfrac{d\left( 2{{e}^{x}}\left( x-1 \right) \right)}{dx}\]
Using the product, we get
\[\Rightarrow \dfrac{d\left( 2{{e}^{x}} \right)}{dx}\left( x-1 \right)+2{{e}^{x}}\dfrac{d\left( x-1 \right)}{dx}\]
Substituting the values of the derivatives, we get
\[\Rightarrow 2{{e}^{x}}\left( x-1 \right)+2{{e}^{x}}\times 1=2x{{e}^{x}}\]
Hence, the derivative of the given expression is \[2x{{e}^{x}}\].

Note: Here we expressed the given expression in the form \[{{e}^{x}}f(x)\]. For the expressions of these forms, we can use a trick for differentiation.
\[\dfrac{d\left( {{e}^{x}}f(x) \right)}{dx}={{e}^{x}}\left( f(x)+\dfrac{d\left( f(x) \right)}{dx} \right)\].
By using this property, we can find the derivative of these types of expression easily.