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

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Solution

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 rule is probably the most widely 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 differentation of $y$ w.r.t. $u$ .
$\frac{d u}{d x} = 2 x$ and $\frac{d y}{d u} = cos u$
and we can write $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ . Hence,
$\frac{d y}{d x} = (cos u) \cdot (2 x) = 2 x \cdot cos (x^{2} - 4)$ .

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