derivative of y=e^(5x)-sin^2(8x)

Question

anonymous · Answer

chain rule.

anonymous · Answer

yes but how do u go about doing it? like which one is the outside eq?

anonymous · Answer

Okay first of all, we need to do each term separately.

anonymous · Answer

$$
e^{5x}
$$Let $u = 5x$.  Now we can say: $$
d(e^{5x})  =d(e^u) = e^u\;du
$$Do you follow so far?

anonymous · Answer

yes

anonymous · Answer

$$
e^udu = e^{5x}d(5x) = e^{5x}5dx
$$

anonymous · Answer

Put it all together, and we get: $$
d(e^{5x}) = 5e^{5x}dx \implies \frac{d(e^{5x})}{dx} = 5e^{5x}
$$

anonymous · Answer

uhh what

anonymous · Answer

ohh kk

anonymous · Answer

Now, do you think you can do the same process for $$
-\sin^2(8x)
$$

anonymous · Answer

In this case, I suggest $u = \sin(8x)$.  And if you have trouble with $du$, then let $v=8x$.

anonymous · Answer

so cosv-sinv?

anonymous · Answer

Okay, first step: $$
d(-\sin^2(8x)) = \ldots
$$What is the first substitution?

anonymous · Answer

What should we choose for $u=f(x)$?

anonymous · Answer

uhh im getting more confused

anonymous · Answer

8x?

anonymous · Answer

Okay... Then $$
d(-\sin^2(8x)) = d(-\sin^2(u))
$$We have mad things a little easier.  What else can we do to make it easier?

anonymous · Answer

change -sin^2 in two parts?

anonymous · Answer

Here is a hint: $$
d(-\sin^2(u)) = d(-[\sin(u)]^2)
$$

anonymous · Answer

-2sin(u)

anonymous · Answer

ok, so $$
d(-\sin^2(u)) = -2\sin(u)d(\sin(u)) = -2\sin(u)\cos(u)du
$$

anonymous · Answer

To make it more clear what I did: $$
d(-\sin^2(u)) = d(-v^2) = -2vdv = -2\sin(u)d(\sin(u)) = -2\sin(u)\cos(u)du
$$

anonymous · Answer

It is a bit trickier when you do it without actually substituting the variable, but it still works!

anonymous · Answer

so the final without subtituting would be 5e^5x-2sin(8x)*cos(sin(8x))?

anonymous · Answer

Hold on...

anonymous · Answer

This is a simple double angle formula:$$
-2\sin(u)\cos(u)du = -\sin(2u)du
$$

anonymous · Answer

Next we have: $$
-\sin(2u)du = -\sin(2(8x))d(8x) = -\sin(16x)8dx 
$$

anonymous · Answer

cause when i attempted, it said 5e^(5x)-16\cos(8x)*\sin(8x)\] was not the answer

anonymous · Answer

The answer is $$
5e^{5x}-8\sin(16x)
$$And if you didn't do the double angle simplification, it would be $$

5e^{5x}-16\sin(8x)\sin(8x)
$$

anonymous · Answer

anonymous · Answer

neither of those were correct

anonymous · Answer

attachment

anonymous · Answer

that was the correct answer^

anonymous · Answer

What was the original problem?

anonymous · Answer

attachment

anonymous · Answer

The problem... is the minus sign.

anonymous · Answer

It was multiplication, not subtraction... that is why we got the wrong answer.