# Thread: Determining Derivative for f(x)=sin(x^3 / (cos(x^3)))

1. ## Determining Derivative for f(x)=sin(x^3 / (cos(x^3)))

For my math assignment I have to find the derivative of the function

f(x)=sin(x^3 / (cos(x^3)))

I know the first step is to apply the chain rule and split up the function into u(x) and v(x) but I always seem to have a hard time determining what u(x) and v(x) are.

I have used https://www.derivative-calculator.net/ which does help me understand a little but I still think it is hard.

2. Originally Posted by BenjaminJosso
For my math assignment I have to find the derivative of the function

f(x)=sin(x^3 / (cos(x^3)))

I know the first step is to apply the chain rule and split up the function into u(x) and v(x) but I always seem to have a hard time determining what u(x) and v(x) are.

I have used https://www.derivative-calculator.net/ which does help me understand a little but I still think it is hard.

$\displaystyle{sin\left [\dfrac{x^3}{cos(x^3)}\right ]}$ ........................ this is what you posted

or

$\displaystyle{\dfrac{sin(x^3)}{cos(x^3)}}$

If you meant the second function, it should have been written as: f(x)=sin(x^3) / cos(x^3)

Please confirm your problem and share any work that you have done (even if you know that there are mistakes).

3. Originally Posted by BenjaminJosso
For my math assignment I have to find the derivative of the function

f(x)=sin(x^3 / (cos(x^3)))
From your grouping symbols, I think you mean the following:

. . . . .$f(x)\, =\, \sin\left(\dfrac{x^3}{\cos(x^3)}\right)$

Originally Posted by BenjaminJosso
I know the first step is to apply the chain rule and split up the function into u(x) and v(x) but I always seem to have a hard time determining what u(x) and v(x) are.
Don't let the book's "formula" confuse you. In practical terms, all the Chain Rule really means is that you should work from the outside in, differentiating layers as you go. In this case:

a. Differentiate the sine, leaving the insides intact. Write a "times" symbol, and

b. differentiate the fraction inside, using the Quotient Rule.

When you're using the Quotient Rule, you'll also be using the Power Rule (for the cubes) and the derivative of cosine. For differentiating the cosine:

c. Differentiate the cosine, leaving the insides intact. Write a "times" symbol, and

d. differentiate the cube inside, using the Power Rule.

That's all there is to it!