Question

Find the derivative of (4+x^2)e^2x w.r to x

Answer 1

Answer:

The derivative of the expression with respect to x is

$\frac{dy}{dx}=2e^{2x}(x^2+x+4)$

Step-by-step explanation:

The given expression is

$y=(4+x^2)e^{2x}$

The derivative of the expression with respect to x is

$\frac{dy}{dx}=\frac{d}{dx}[(4+x^2)e^{2x}]$

This can be solved using the product rule which is $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

Therefore, we write

$\frac{d}{dx}[(4+x^2)e^{2x}]=(4+x^2)\frac{d}{dx}(e^{2x})+e^{2x}\frac{d}{dx}(4+x^2)\\=(4+x^2)(2e^{2x})+e^{2x}(2x)\\=2e^{2x}(x^2+x+4)$

Therefore,

The derivative of the expression with respect to x is

$\frac{dy}{dx}=2e^{2x}(x^2+x+4)$

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Answer 2

Answer:

The derivative of the function $(4 + x^2)e^{2x}$ with respect to $x$ is $2e^{2x}(8+x+x^2)$.

Step-by-step explanation:

Product rule for differentiation:-

The differentiation of the function of the form u(x)v(x) is,

$\frac{d}{dx}(uv)=u\frac{d}{dx}(v)+v\frac{d}{dx}(u)$

According to the question,

Given:-

The function is $(4 + x^2)e^{2x}$

To find:-

The derivative of the function $(4 + x^2)e^{2x}$ with respect to $x$.

Consider the given function as follows:

$y=(4 + x^2)e^{2x}$ . . . . . (1)

Rewrite the function (1) as follows:

$y=4e^{2x} + x^2e^{2x}$

Differentiate both the sides with respect to $x$ as follows:

$\frac{d}{dx}( y)=\frac{d}{dx}(4e^{2x}) +\frac{d}{dx}( x^2e^{2x})$

$\frac{d}{dx}( y)=4\frac{d}{dx}(e^{2x}) +\frac{d}{dx}( x^2e^{2x})$

Using the product rule, simplify the derivative as follows:

$\frac{dy}{dx}=4\frac{d}{dx}(e^{2x}) +x^2\frac{d}{dx}(e^{2x})+e^{2x}\frac{d}{dx}(x^2)$

    $=4(2e^{2x}) +x^2(2e^{2x})+e^{2x}(2x)$

$\frac{dy}{dx}=8e^{2x}+2x^2e^{2x}+2xe^{2x}$

Now,

Take the term 2$e^{2x}$ common out as follows:

$\frac{dy}{dx}=2e^{2x}(8+x^2+x)$

$\frac{dy}{dx}=2e^{2x}(8+x+x^2)$

Final answer: The derivative of the function $(4 + x^2)e^{2x}$ with respect to $x$ is $2e^{2x}(8+x+x^2)$.

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