Find the derivative of `y=sin(x^2-4)`.

Options

A. `2xcos(x^2-4)`
B. `2xsin(x^2-4)`
C. `xcos(x^2-4)`
D. `2xcos(x^2+4)`

Answer

A

Explanation

We now know how to differentiate `sinx` and `x^2-4`, but how do we differentiate a composite like `sin(x^2-4)`? The answer is, with the chain rule, which says that the derivative of the composite of two differentiable functions is the product of their derivatives evaluated at appropriate points. The chain rule is probably the most widely used differentiation rule in mathematics.
Let `u=x^2-4`
Then `y=sinu`
We know the differentiation of u w.r.t. x and differentiation of y w.r.t.u.
`(du)/(dx)=2x` and `(dy)/(du)=cosu`
and we can write `(dy)/(dx)=(dy)/(du)*(du)/(dx)`.
Hence, `(dy)/(dx)=(cosu)*(2x)=2x*cos(x^2-4)`