Question

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

• A. $2xcos(x^2-4)$
• B. $2xcos{x}^2$
• C. $xcos(x^2-4)$
• D. none of these

Answer 1

The correct option is A

$2xcos(x^2-4)$

We now know how to differentiate sin x 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 derivatives of the composite of two differentiable fuctions is the product of their derivatives evaluated at appropriate points. The Chain Rule is probably the most used differentiation rule in mathematics.

Let $u=x^2-4$
Then y=sin u
We know the differentiation of u w.r.t x and differentiation of y w.r.t u.
$\frac{du}{dx}=2x\&\frac{dy}{du}=cosu$ we can write $\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}$
$\frac{dy}{dx}=\left(cosu\right).\left(2x\right)=2x.cos(x^2-4)$