Find the derivative of sin (3x + 2) w.r.t. x from first principle… - Vidyadip

Question

Find the derivative of sin (3x + 2) w.r.t. x from first principle.

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Answer

$\frac{\Delta\text{y}}{\Delta\text{x}}=\frac{\sin(\text{3x + 3} \Delta\text{x}+2)-\sin(\text{3x + 2})}{\Delta\text{x}}$
$\therefore\frac{\text{dy}}{\text{dx}}=\lim_{\Delta\text{x} \to 0}\cdot\frac{2\cos\Bigg[\text{3x + 2 +}\frac{3\Delta\text{x}}{2}\Bigg]\sin\Bigg(3\frac{\Delta\text{x}}{2}\Bigg)}{\Delta\text{x}}$
$=2\cos(\text{ 3x + 2})\cdot\lim_{\Delta\text{x} \to 0}\frac{\sin\Big(3\frac{\Delta\text{x}}{2}\Big)}{3\frac{\Delta\text{x}}{2}}\cdot\frac{3}{2}$
= 3 cos (3x + 2).

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