Question

???

Answer 1

\(f(x) = (x+2)^{7x}\) , right?

Answer 2

Yes

Answer 3

so, \(y= (x+2)^{7x}\)
ln both sides, what do you get?

Answer 4

What do you mean?

Answer 5

\(ln\), or log .

Answer 6

I did chain rule and got 7x(x+2)^(6x)

Answer 7

No way!! this is an exponent function with variable on the exponent.
You use chain rule if the exponent is a constant.
Only one way to take the exponent down is take \(ln\) both sides and use implicit derivative to find dy/dx.

Answer 8

Im not sure how to do that

Answer 9

ok, I do it for you as sample
\(ln y = 7x ln(x+2)\)
Now take derivative both sides
\(\dfrac{y'}{y}= 7ln(x+2) +\dfrac{7x}{x+2}\)
multiple y both sides, and replace \(y = (x+2)^{7x}\) , you get the answer

Answer 10

Okay so it would be (7ln(x+2) + 7x/(x+2))(x+2)^7x

Answer 11

yes

Answer 12

Thank you!!

Answer 13

oops used implicit differentiation

another way is to use this factoid (by definition):
\[ x= e^{\ln x } \]
use that to say
\[ (x+2)= e^{\ln(x+2)} \]
and
\[ (x+2)^{7x}= \left(e^{\ln(x+2)} \right)^{7x} \\y= e^{7x\ln(x+2)} \]

now you can take the deriviative

Answer 14

Thank you so much @phi !!!