Derivative of $f (x) = sin (x^{3} + 2 x + 1)$ is

cos (3 x^{2} + 2)

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cos (x^{3} + 2 x + 1)

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cos (x^{3} + 2 x + 1) (3 x^{2} + 2)

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cos (x^{3} + 2 x + 1) (3 x^{2} - 2)

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Solution

The correct option is C $cos (x^{3} + 2 x + 1) (3 x^{2} + 2)$
Given, $f (x) = sin (x^{3} + 2 x + 1)$
So, the derivative of the function is
$\frac{d f (x)}{d x} = \frac{d (sin (x^{3} + 2 x + 1))}{d x}$
Using chain rule we have
$\frac{d (sin (x^{3} + 2 x + 1))}{d (x^{3} + 2 x + 1)} \times \frac{d (x^{3} + 2 x + 1)}{d x}$
$\Rightarrow \frac{d (sin (x^{3} + 2 x + 1))}{d (x^{3} + 2 x + 1)} \times [\frac{d x^{3}}{d x} + 2 \frac{d x}{d x} + \frac{d 1}{d x}]$
$\Rightarrow cos (x^{3} + 2 x + 1) (3 x^{2} + 2)$

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