Question

How do you find the derivative of `y= a^3 + cos^3 x`?

Answer 1

`-3sinxcos^2x`

Explanation:

Here the variable is x and a is a constant.

`color(orange)"Reminder:"d/dx"(constant)"=0`

Express `cos^3x=(cosx)^3`

and differentiate using the `color(blue)"chain rule and power rule"`

`d/dx(f(g(x)))=f'(g(x)).g'(x)........ (A)`
`"----------------------------------------------------"`
`f(g(x))=(cosx)^3rArrf'(g(x))=3(cosx)^2=3cos^2x`

`g(x)=cosxrArrg'(x)=-sinx`
`"----------------------------------------------------"`
Substitute these values into(A)

`rArr3cos^2x(-sinx)=-3sinxcos^2x`

So y=`a^3+cos^3x`

`rArrdy/dx=0-3sinxcos^2x=-3sinxcos^2x`