Question

Derivative 3/(5x^2+sin2x)^3/2

Answer 1

This?:
\[y=\frac{3}{(5x^2+\sin2x)^{3/2}}\]

Answer 2

yes

Answer 3

Well, the first thing we would want to do is move that whole denominator into the numerator by making the exponent negative.

Answer 4

I would say use the quotient rule, but I'll wait for Psymon

Answer 5

No need for quotient rule :P Rewrite:

\[3(5x ^{2}+\sin2x)^{-\frac{ 3 }{ 2 }}\]
I think doing it this way makes things way easier.

Answer 6

Ofc, there goes Psymon making me feel bad ;-;

Answer 7

Sorry D:

Answer 8

Just give up, Luigi.

Answer 9

Alright, so this is a simple power rule, but with a chain rule. Chain rule is when we have functions that come in layers. In your case we have two layers. Or if you wish, an outer function and an inner function. Here are the outer and inner portions:
\[3(-----)^{-\frac{ 3 }{ 2 }}\]I just write it like this to emphasize the portion of the function we are actually dealing with. The inner function is of course:
\[5x ^{2}+\sin2x \]
Now when we do chain rule, we take the derivative of each layer or each inner function and then multiply the results. So what we will be doing is taking the derivative of:
\[3(----)^{-\frac{ 3 }{ 2 }}\] and then multiplying it by the derivative of
\[5x ^{2}+\sin2x\]
So for the first portion, this is just a simple power rule. If you recall, we bring the power down as a multiplication, then lower the power by 1. You think you can do that part?

Answer 10

\[y=\frac{3}{(5x^2+\sin2x)^{ 3 \over 2}}\]
\[\frac{dy}{dx}=\frac{d}{dx}\frac{3}{(5x^2+\sin2x)^{ 3 \over 2}}\]
\[\frac{dy}{dx}=\frac{(5x^2+\sin2x)^{ 3 \over 2}\frac{d}{dx}3-3\frac{d}{dx}(5x^2+\sin2x)^{ 3 \over 2}}{[(5x^2+\sin2x)^{ 3 \over 2}]^2}\]
\[\frac{dy}{dx}=\frac{(5x^2+\sin2x)^{ 3 \over 2} \times 0-3 \times \frac{3}{2}(5x^2+\sin2x)^{ 1 \over 2}(10x+2 \cos2x)}{[(5x^2+\sin2x)^{ 3 }]}\]
\[=\frac{0-\frac{9}{2}(5x^2+\sin 2x)^{1 \over 2}(10x+2\cos2x)}{(5x^2+\sin 2x)^{3}}\]
\[=\frac{\frac{9}{2}(10x+2\cos2x)}{(5x^2+\sin 2x)^{3 -\frac{1}{2}}}\]
\[=\frac{9(10x+2\cos2x)}{2(5x^2+\sin 2x)^{\frac{5}{2}}}\]

Answer 11

I would use logarithmic differentiation.

Answer 12

Ln both sides and use log properties to make the derivative very very easy.

Answer 13

I dont think id do logarithmic personally, lol.