Differentiate the following functions with respect to x: 3x^2 7-x… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\frac{3\text{x}^2\sin\text{x}}{\sqrt{7-\text{x}^2}}$

✓

Answer

Let $\text{y}=\frac{3\text{x}^2\sin\text{x}}{\sqrt{7-\text{x}^2}}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\bigg\{\frac{3\text{x}^2\sin\text{x}}{(\sqrt{7-\text{x}^2})^\frac{1}{2}}\bigg\}$
$=\frac{(7-\text{x}^2)^\frac{1}{2}\times\frac{\text{d}}{\text{dx}}(3\text{x}^3\sin\text{x})-(3\text{x}^2\sin\text{x})\frac{\text{d}}{\text{dx}}(7-\text{x}^2)^\frac{1}{2}}{\Big[(7-\text{x}^2)^\frac{1}{2}\Big]^2}$
[Using quotient rule, chain rule and product rule]
$\Bigg[\frac{(7-\text{x}^2)^\frac{1}{2}\times3\Big[\text{x}^2\frac{\text{d}}{\text{dx}}(\sin\text{x})+\sin\text{x}\frac{\text{d}}{\text{dx}}(\text{x}^2)\Big]-3\text{x}^2\sin\text{x}\times\frac{1}{2}(7-\text{x}^2)\times\frac{\text{d}}{\text{dx}}(7-\text{x}^2)}{(7-\text{x}^2)}\Bigg]$
$\Bigg[\frac{(7-\text{x}^2)^\frac{1}{2}3(\text{x}^2\cos\text{x}+2\text{x}\sin\text{x})-3\text{x}^2\sin\text{x}\times\frac{1}{2}(7-\text{x}^2)^\frac{-1}{2}(-2\text{x})}{(7-\text{x}^2)}\Bigg]$
$=\Bigg[\frac{(7-\text{x}^2)^\frac{1}{2}\times3(\text{x}^2\cos+2\text{x}\sin\text{x})}{(7-\text{x}^2)}+\frac{3\text{x}^2\sin\text{x}(7-\text{x}^2)^\frac{-1}{2}}{(7-\text{x}^2)}\Bigg]$
$\bigg[\frac{6\text{x}\sin\text{x}+3\text{x}^2\cos\text{x}}{\sqrt{(7-\text{x}^2)}}+\frac{3\text{x}^3\sin\text{x}}{(7-\text{x}^2)^\frac{3}{2}}\bigg]$
So,
$\frac{\text{d}}{\text{dx}}\Big(\frac{3\text{x}^2\sin\text{x}}{\sqrt{7-\text{x}^2}}\Big)\bigg[\frac{6\text{x}\sin\text{x}+3\text{x}^2\cos\text{x}}{\sqrt{(7-\text{x}^2)}}+\frac{3\text{x}^3\sin\text{x}}{(7-\text{x}^2)^\frac{3}{2}}\bigg]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If a young man rides his motorcycle at 25 km/hour, he had to spend Rs. 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as an LPP and solve it graphically.

If x and y are connected parametrically by the equations given in Exercise without eliminating the parameter, Find $\frac{\text{dy}}{\text{dx}}.$
$\text{x}=\frac{\sin^3\text{t}}{\sqrt{\cos2\text{t}}},\text{y}\frac{\cos^3\text{t}}{\sqrt{\cos2\text{t}}}$

Evaluate the following integrals:
$\int\limits^\frac{\pi}{2}_0\text{x}^2\sin\text{x dx}$

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is:

Symmetric and transitive only.
Symmetric only.
Transitive only.
None of these.

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

Evalute the following integrals:
$\int\frac{1-\cot\text{x}}{1+\cot\text{x}}\text{dx}$

In a shop X, 30 tins of pure ghee and 40 tins of adulterated ghee which look alike, are kept for sale while in shop Y, similar 50 tins of pure ghee and 60 tins of adulterated ghee are there. One tin of ghee is purchased from one of the randomly selected shops and is found to be adulterated. Find the probability that it is purchased from shop Y. What measures should be taken to stop adulteration?

Find the equation of the plane passing through the line of intersection of the planes $\vec{\text{r}}.\Big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\Big)=1\ \text{and}\ \vec{\text{r}}.\Big(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}\Big)+4=0$ and parallel to x-axis.

Find the real values of $\lambda$ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions:
$2\lambda\text{x}-2\text{y}+3\text{z}=0,$
$\text{x}+\lambda\text{y}+2\text{z}=0,$
$2\text{x}+0\text{y}+\lambda\text{z}=0$

Let $\vec{\text{a}}=\text{x}^2\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\text{x}^2\hat{\text{i}}+5\hat{\text{j}}-4\hat{\text{k}}$ be three vectors. Find the valuse of x for which the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ is acute and the angle between $\vec{\text{b}}$ and $\vec{\text{c}}$ is obtuse.