Question

differentiate cos²(x³) with respect to x³​

Answer 1

Answer:

-sin(2x³)

Step-by-step explanation:

let x³ = y

so, $\frac{d}{dx^3} (cos^2(x^3)) = \frac{d}{dy}(cos^2y)$

= 2cosy(-siny)

= -2sinycosy

= -sin2y

= -sin2x³                  (as,y=x³)

Answer 2

Step-by-step explanation:

It could be differentiated using chain rule only

cos2(x3)=(cos(x3))2

usually we would say

(f(g(x)))'=f'(g(x))⋅g'(x)

but this time we have

(cos(x3))2=f(g(h(x)))

where f(x)=x2, g(x)=cos(x) and h(x)=x3

It turns out that the formula above is kind of "recursive". It means that if there is more functions inside, we can plug g(h(x)) as second function ant then find derivative of it by chain rule:

(f(g(h(x))))'=f'(g(h(x)))⋅(g(h(x)))'=

=f'(g(h(x)))⋅g'(h(x))⋅h'(x)

(That's the essence of "chain" rule)

For our example

(cos(x3))2