anonymous one year ago ???

1. anonymous

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

2. anonymous

Yes

3. anonymous

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

4. anonymous

What do you mean?

5. anonymous

$$ln$$, or log .

6. anonymous

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

7. anonymous

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.

8. anonymous

Im not sure how to do that

9. anonymous

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

10. anonymous

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

11. phi

yes

12. anonymous

Thank you!!

13. phi

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

14. anonymous

Thank you so much @phi !!!