What is the value of ddx ( x^3 ⁡ x^2)? - Vidyadip

MCQ

What is the value of $\text{ddx} (\sin x^3 \cos⁡ x^2)$?

A
$ 3 x^2 \cos x^2 \cos x^3+2 x \sin x^3 \sin x^2 $
✓
$ 3 x^2 \cos 2 \cos x^3-2 x \sin x^3 \sin x^2 $
C
$ 2 x \cos x^2 \cos x^3-2 x \sin x^3 \sin x^2 $
D
$ 2 x \cos x^2 \cos x^3+3 x^2 \sin x^3 \sin x^2 $

✓

Answer

Correct option: B.

$ 3 x^2 \cos 2 \cos x^3-2 x \sin x^3 \sin x^2 $

We follow product rule $\frac{\text{d}}{\text{dx}}(\text{f}.\text{g}.)=\text{g}.\frac{\text{d}}{\text{dx}}(\text{f})+\text{f}.\frac{\text{dy}}{\text{dx}}(\text{g})$
Here $ \text{f} = \sin⁡ x^3$ and $g = \cos⁡ x^2$
$\frac{\text{d}}{\text{dx}} (\text{f}) = 3x^2 \cos⁡ x^3$
$\frac{\text{d}}{\text{dx}} (\text{g}) = -2\text{x}\ \sin \text{x}^2$
We now substitute this in our main equation,
$=\cos⁡ x^2.3x^2 \cos⁡ x^3 + \sin⁡ x^3.(-2x \sin x^2)$
$=3x^2 \cos x^2 \cos⁡ x^3 – 2x \sin⁡ x^3 \sin x2$

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