Question

d/dx [tan^3(x)]

Answer 1

You understand that the notation is for a derivative right? What rules do you think you should use?

Answer 2

the power rule

Answer 3

How can you rewrite tan^3(x)? Then you might recognize another rule.

Answer 4

i have no clue

Answer 5

Have you heard of the chain rule?

Answer 6

yeah

Answer 7

You can rewrite tan^3(x) as (tan(x))^3.

Answer 8

oh ok

Answer 9

then what do you do?

Answer 10

You can rewrite (tan(x))^3 as (sinx/cosx)^3 and use the chain and quotient rules. Otherwise you would use the chain rule outright and use the derivative of the tangent function, which you should memorize. If you don't know it then you could google it to jog your memory.

Answer 11

can you walk me through how to do it either way?

Answer 12

Alright, so we start with using the chain rule, in which we will use the equation (tanx)^3. We change it to (sinx/cosx)^3 = ?

To use the chain rule, remember to make the inside BLOP, and take the derivative of the outside first. So we first make tanx into BLOP!
(BLOP)^3
3(BLOP)^2
But we are still missing something from the chain rule, which states we have to multiply now the 3(BLOP)^2 by the derivative of BLOP.

So now we have 3(BLOP)^2 dBLOP/dx.

We know BLOP = tanx = sinx/cosx.

If we use sinx/cosx then we need to use our quotient rule. Can you finish it now?

Answer 13

Remember when you take the derivative of tanx = sinx/cosx that we have another blop, which is now the x.

Answer 14

what is blop?

Answer 15

BLOP could be anything. It's just like saying your putting something over that area of the equation and ignoring it for the time being. Like saying the derivative of x^2

dBLOP^2/dx = 2BLOP*DBLOP/dx = 2BLOP = 2x.

Answer 16

Do you know how to use the quotient rule and the derivatives of sinx and cosx?

Answer 17

sometimes they call the chain rule the general power rule:

\[\large \frac{d}{dx}tan^3x=\frac{d}{dx}[tanx]^3=3(tanx)^{(3-1)}\cdot [tanx]' \]

can you take it from here?

Answer 18

what more do i need to do?

Answer 19

that last part [tanx]'...

Answer 20

You still need to find the derivative of the tanx.

Answer 21

secx^2

Answer 22

Exactly.

Answer 23

great now replace that with [tanx]' and ur done...:)

Answer 24

Now put it all together.

Answer 25

yay!!

Answer 26

Remember too that if you have something other than x behind the tanx, you would have to continue your chain rule. Luckily this time x' = 1.

Answer 27

i have another that i am posting and help would be greatly appreciated!