Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Differentiate of the following w.r.t. x:
$\sin^\text{m}\text{x}\cdot\cos^\text{n}\text{x}$
✓

Answer

Let $\text{y}=\sin^\text{m}\text{x}\cdot\cos^\text{n}\text{x}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big[(\sin\text{x})^\text{m}\cdot(\cos\text{x})^\text{n}\big]$
$=(\sin\text{x})^\text{m}\cdot\frac{\text{d}}{\text{dx}}(\cos\text{x})^\text{n}+(\cos\text{x})^\text{n}\cdot\frac{\text{d}}{\text{dx}}(\sin\text{x})^\text{m}$
$=(\sin\text{x})^\text{m}\cdot\text{n}(\cos\text{x})^{\text{n}-1}\cdot\frac{\text{d}}{\text{dx}}\cos\text{x}+(\cos\text{x})^\text{n}\text{m}(\sin\text{x})^{\text{m}-1}\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}$
$=(\sin\text{x})^\text{m}\cdot\text{n}(\cos\text{x})^{\text{n}-1}(-\sin\text{x})+(\cos\text{x})^\text{n}\cdot\text{m}(\sin\text{x})^{\text{m}-1}\cos\text{x}$
$=-\text{n}\cdot\sin^\text{m}\text{x}\cdot\sin\text{x}\cdot\cos^\text{n}\text{x}\cdot\frac{1}{\cos\text{x}}+\text{m}\cdot\sin^\text{m}\text{x}\cdot\frac{1}{\sin\text{x}}\cdot\cos^\text{n}\text{x}\cdot\cos\text{x}$
$=-\text{n}\sin^\text{m}\text{x}\cdot\cos^\text{n}\text{x}\cdot\tan\text{x}+\text{m}\sin^\text{m}\text{x}\cdot\cos^\text{n}\text{x}\cdot\cot\text{x}$
$=\sin^\text{m}\text{x}\cdot\cos^\text{n}\text{x}\big[\text{m}\cot\text{x}-\text{n}\tan\text{x}\big]$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If function $F ( x )=\left\{\begin{array}{l}k x+1, x \leq \pi \\ \cos x, x>\pi\end{array}\right.$ is continuous at $x=\pi$ then find $k$.
→
Find $|\vec{\text{a}|}$ and $\big|\vec{\text{b}}\big|$ if
$\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=3$ and $|\vec{\text{a}}|=2\big|\vec{\text{b}}\big|$
→
Solve the following differential equation:$2 xydx + (x^{2} +2y^2) dy = 0$
→
Evaluate the following integrals:$\int\text{e}^{\text{x}}\big[\sec\text{x}+\log(\sec\text{x}+\tan\text{x})\big]\text{dx}$
→
If $\text{A}=\begin{bmatrix}-1 & -2 & -2 \\2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix},$ show that $A = 3A^T$.
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}'(\text{x})=\frac{\text{x}}{2}+\frac{2}{\text{x}}, \text{x}\geq0$
→
Evaluate the following integrals:
$\int\frac{1-\cos\text{x}}{1+\cos\text{x}}\text{dx}$
→
If $\text{A}=\text{diag}\begin{pmatrix}2&-5&9\end{pmatrix},\text{ B}=\text{diag}\begin{pmatrix}1&1&-4\end{pmatrix}$ and $\text{C}=\text{diag}\begin{pmatrix}-6&3&4\end{pmatrix},$ find.
$\text{B}+\text{C}-2\text{A}$
→
 Let $A=R_0 \times R$, where $R_0$ denote the set of all non-zero real numbers. $A$ binary operation ' $\odot$ ' is defined on $A$ as follows: $(a, b) \odot(c, d)=(a c, b c+d)$ for all $(a, b),(c, d) \in R_0 \times R$.
Find the identity element in A .
→
Obtain the equation of the plane passing through the point (1, - 3, -2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.
→