How do you differentiate # f(x)= (7e^x+x)^2 # using the chain rule.?

1 Answer
Jacob F.
Dec 18, 2015

#f'(x) = 2(7e^x + x)(7e^x + 1)#

Explanation:

With the chain rule, if you have some composition of functions that looks like

#f(x) = g(h(x))#,

then the derivative #f'(x)# is equal to

#f'(x) = d/(dh)g(h)*d/(dx)h(x)#.

Essentially, differentiate the outside function while treating the whole inside function as if it's a single variable, and multiply it by the derivative of the inside function.

To illustrate what I mean, just imagine #h = 7e^x + x# for a second.

Then we have

#f(x) = g(h(x)) = h^2#.

Right? #g(h)# is just all of #h# squared, and so is #f(x)#.

So, to find the derivative, we'll just apply the formula we have above.

#f'(x) = d/(dh)g(h)*d/(dx)h(x)#

The derivative of #g(h) = h^2# with respect to #h# is just #2h#. (power rule)

And, the derivative of #h# with respect to #x# is #h' = 7e^x + 1#.

Let's plug all that in:

#f'(x) = 2h*h'#

#-> f'(x) = 2(7e^x + x)(7e^x + 1)#

And there's our answer.

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