How do you differentiate #f(x)=e^x(x+2)^2# using the product rule?

Question

Isaiah Allen · Accepted Answer

#f'(x)=e^x(x+2)(x+4)#

In line with the product rule:

#f'(x)=(x+2)^2d/dx[e^x]+e^xd/dx[(x+2)^2]#

Determine every derivative.

#d/dx[e^x]=e^x#

The chain rule will be applied to the differentiation that follows:

#d/dx[(x+2)^2]=2(x+2)d/dx[x+2]=2(x+2)=2x+4#

Re-plug these in.

#f'(x)=e^x(x+2)^2+e^x(2x+4)#

#f'(x)=e^x(x^2+4x+4+2x+4)#

#f'(x)=e^x(x^2+6x+8)#

#f'(x)=e^x(x+2)(x+4)#

Adam Adkins · Answer

undefined To differentiate $ f(x) = e^x(x+2)^2 $ using the product rule:

1. Identify the two functions being multiplied together: $ e^x $ and $ (x+2)^2 $.
2. Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
3. Compute the derivative of $ e^x $ which is $ e^x $.
4. Compute the derivative of $ (x+2)^2 $ using the chain rule, which is $ 2(x+2) $.
5. Apply the product rule formula:

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

6. Simplify the expression:

$$
= e^x(x+2)^2 + 2e^x(x+2)
$$

7. This is the derivative of $ f(x) $ with respect to $ x $.