Differentiate the following w.r.t. x: ^2+ ^2x+ ^2(x^2) - Vidyadip

Question

Differentiate the following w.r.t. x:
$\sin\text{x}^2+\sin^2\text{x}+\sin^2(\text{x}^2)$

✓

Answer

Let $\text{y}=\sin\text{x}^2+\sin^2\text{x}+\sin^2(\text{x}^2)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\sin\big(\text{x}^2\big)+\frac{\text{d}}{\text{dx}}\big(\sin\text{x}\big)^2+\frac{\text{d}}{\text{dx}}(\sin\text{x}^2)^2$
$=\cos\big(\text{x}^2\big)\frac{\text{d}}{\text{dx}}\big(\text{x}^2\big)+2\sin\text{x}\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}+2\sin^2\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}^2$
$=2\text{x}\cos\text{x}^2+2\cdot\sin\text{x}\cdot\cos\text{x}+2\sin\text{x}^2\cos\text{x}^2\cdot\frac{\text{d}}{\text{dx}}\text{x}^2$
$=2\text{x}\cos\text{x}^2+2\cdot\sin\text{x}\cdot\cos\text{x}+2\sin\text{x}^2\cos\text{x}^2\cdot2\text{x}$
$=2\text{x}\cos\text{x}^2+\sin2\text{x}+\sin\big(2\text{x}^2\big)\cdot2\text{x}$
$=2\text{x}\cos\text{x}^2+2\text{x}\cdot\sin2\big(\text{x}^2\big)+\sin2\text{x}$

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 "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\tan^{-1} \frac{\text{x - 3}}{\text{x - 4}} + \tan^{-1} \frac{\text{x + 3}}{\text{x + 4}} = \frac{\pi}{4},$ then find the value of x.

Check the commutativity and associativity of the following binary operations:
'o' on Q defined by $\text{a o b}=\frac{\text{ab}}{2}$ for all a, b ∈ Q.

Show that $\big|\vec{a}|\ \vec{b}+\big|\vec{b}\big|\ \vec{a}$ is perpendicular to $\big|\vec{a}|\ \vec{b}-\big|\vec{b}\big|\ \vec{a},$ for any two nonzero vectors $\vec{a}\ \text{and}\ \vec{b}.$

Find the cartesian equation of the line which passes through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{\text{x}+3}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+8}{6}.$

Show that $2\tan^{-1}(-3)=\frac{-\pi}{2}+\tan^{-1}\Big(\frac{-4}{3}\Big).$

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}(\text{x}-\text{a}){\sin}\Big(\frac{1}{\text{x}-\text{a}}\Big) & \text{x} \neq \text{a}\\\ 0, & \text{ x} = \text{a}\end{cases}\text{at x}=\text{a}$

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}.$

For the principal values, evaluate the following:
$\sin^{-1}[\cos\{2\text{cosec}^{-1}(-2)\}]$

Evaluate the following integrals:
$\int_{2}^\limits{4}\frac{\text{x}}{\text{x}^2+1}\text{ dx}$

Solve the following:
$\cos^{-1}\text{x}+\sin^{-1}\frac{\text{x}}{2}=\frac{\pi}{6}$