## lovesit2x 2 years ago Differentiate y = e^(4x) cos x with respect to x dy/dx = ?

1. zepdrix

Hmm looks like we'll have to apply the product rule :)

2. Xavier

You have a product so use the product rule

3. zepdrix

$\large y=e^{4x}\cos x$$\large y'=\color{royalblue}{\left(e^{4x}\right)'}\cos x+e^{4x}\color{royalblue}{\left(\cos x\right)'}$ Understand the setup for Product Rule? We have to take the derivative of the blue terms.

4. lovesit2x

Yes

5. zepdrix

So what's the derivative of cos x? :)

6. lovesit2x

sin x ?

7. lovesit2x

wait -sinx

8. zepdrix

Yes good c:$\large y'=\color{royalblue}{\left(e^{4x}\right)'}\cos x+e^{4x}\left(-\sin x\right)$

9. zepdrix

Do you remember the derivative of $$\huge e^x$$ ?

10. lovesit2x

Nope :)

11. zepdrix

Oooo tsk tsk! That's a fun easy one that you'll want to remember! c: $\huge \left(e^x\right)'=e^x$ It gives us the same thing back. This same thing will happen in our problem here, except we'll have an extra step. Since our exponent is more than just $$\large x$$, we have to apply the chain rule. Multiply the result by the derivative of the exponent. $\huge \left(e^{4x}\right)'=e^{4x}\left(4x\right)'$

12. zepdrix

Hmm so what is the derivative of that exponent. The derivative of $$\large 4x$$.... hmmmm

13. lovesit2x

4x

14. zepdrix

The derivative of 4x is 4x? Hmm no that's not going to work :c

15. lovesit2x

4

16. lovesit2x

lol

17. zepdrix

Yessss good :) Giving us, $$\huge e^{4x}(4)$$ Which gives us an answer of,$\huge y'=4e^{4x}\cos x+e^{4x}\left(-\sin x\right)$

18. lovesit2x

WOW...you just made math fun. lol Thanks

19. zepdrix

lol :3 np

20. lovesit2x

wait...its with respect to x