Where did the cos7x come from in the derivative of y=(1+cos²7x)³?

[ms. confused]

OK, in Calculus, in finding the derivative of y=(1+cos²7x)³ my textbook says the answer is -42(1+cos²7x)²cos7xsin7x. Where did they get the cos7x from?

 

[Doc Al]

Hint: What's the derivative of $cos^2(7x)$ ?

 

[dextercioby]

What is the derivative of

$$1+\cos^{2}7x$$

?

Daniel.

 

[chroot]

The derivative of

$$\left( 1 + \cos^2 7x \right) ^3$$

is

$$3 \cdot \left( 1 + \cos^2 7x \right) ^2$$

...times the derivative of the stuff inside the ^3, which is

$$2 \cdot \cos 7x$$

...times the derivative of the stuff inside the ^2, which is

$$-\sin 7x$$

...times the derivative of the stuff inside the sin, which is 7.

So, in total, we have

$$3 \cdot \left( 1 + \cos^2 7x \right) ^2 \cdot 2 \cdot \cos 7x \cdot -\sin 7x \cdot 7$$

which reduces to the answer you've been shown.

- Warren

 

[dextercioby]

I think Warren started writing his post ahead of us,Doc.

Daniel.

 

[ms. confused]

Alright, I think I was just overlooking a few things. Thank-you all for pointing me in the right direction.