Question

What is the derivative of `cos(a^3+x^3)`?

Answer 1

`-(3x^2)(sin(a^3 + x^3))`

Explanation:

Use chain rule for derivatives.

Consider this as `d/dx (cos(f(x)))` where `f(x)` = `a^3 + x^3`

The answer would be composition of derivatives of `cos(x)` (and putting x as f(x) after differentiating and f(x). Let me demonstrate this in this question.

Derivative of `cos(x)` is `-sin(x)`. Now, let's substitute x with f(x)

So the answer is `(-sin(f(x))xx(d/dx(f(x))))`.

Now, `f(x) = a^3 + x^3`. Assuming a to be a constant, `a^3` is a constant and derivative of a constant is `0`. Derivative of `x^3` is `3x^2`. I won't explain this because you need to learn this yourself if you can't already figure it out.

So back to the answer.

Ans: `(-sin(a^3 + x^3)xx(d/dx(a^3 + x^3)))`

Final Ans: `-3x^2xxsin(a^3 + x^3)`

Answer 2

Just another way of saying the same thing

`=>(dy)/(dx)=-3x^2sin(a^3+x^3)`

Explanation:

Let `u=a^3+x^3" "->" "(du)/(dx)=3x^2`

Let `y=cos(u)" "->" "(dy)/(du)=-sin(u)`

But `(dy)/(dx)=(du)/(dx)xx(dy)/(du)`

`=>(dy)/(dx)=-3x^2sin(a^3+x^3)`

Answer 3

`d/dx=-sin(a^3+x^3)3x^2`

Explanation:

the main function is `cos(x)`
the sub function is `a^3+x^3`
by the chain rule
main function should differentiate first ，and then differentiate sub function
so
`d/dx=-sin(a^3+x^3)3x^2`