Differentiate following w.r.t. x: ^n (ax^2+bx+c ) - Vidyadip

Question

Differentiate following w.r.t. x:
$\sin^\text{n}\big(\text{ax}^2+\text{bx}+\text{c}\big)$

✓

Answer

Let $\text{y}=\sin^\text{n}\big(\text{ax}^2+\text{bx}+\text{c}\big)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big[\sin\big(\text{ax}^2+\text{bx}+\text{c}\big)\Big]^\text{n}$
$=\text{n}\cdot\Big[\sin\big(\text{ax}^2+\text{bx}+\text{c}\big)\Big]^{\text{n}-1}\cdot\frac{\text{d}}{\text{dx}}\sin\big(\text{ax}^2+\text{bx}+\text{c}\big)$
$=\text{n}\cdot\sin^{\text{n}-1}\big(\text{ax}^2+\text{bx}+\text{c}\big)\cdot\cos\big(\text{ax}^2+\text{bx}+\text{c}\big)\cdot\frac{\text{d}}{\text{dx}}\big(\text{ax}^2+\text{bx}+\text{c}\big)$
$=\text{n}\cdot\sin^{\text{n}-1}\big(\text{ax}^2+\text{bx}+\text{c}\big)\cdot\cos\big(\text{ax}^2+\text{bx}+\text{c}\big)\cdot(2\text{ax + b})$
$=\text{n}\cdot(2\text{ax + b})\cdot\sin^{\text{n}-1}\big(\text{ax}^2+\text{bx}+\text{c}\big)\cdot\cos\big(\text{ax}^2+\text{bx}+\text{c}\big)$

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" 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

Verify Mean Value Theorem, if f(x) = x³ – 5x² – 3x in the interval [a, b], where a = 1 and b = 3. Find all $\text{c}\in(1,\ 3)$ for which f′(c) = 0.

Differentiate the following w.r.t. x:
$\frac{8^\text{x}}{\text{x}^8}$

The probability is 0.02 that an item produced by a factory is defective. A shipment of 10,000 items is sent to its warehouse. Find the expected number of defective items and the standard deviation.

If $\log\text{y}=\tan^{-1}$ show that $(1+\text{x}^2)\text{y}_2+(2\text{x}-1)\text{y}_1=0$

Prove that:
$\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{8}=\frac{\pi}{4}$

Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3cm and 3.0005cm, respectively.

Find matrix $\text{A},\text{if}\begin{bmatrix}1&2&-1\\0&4&9\end{bmatrix}+\text{A}=\begin{bmatrix}9&-1&4\\-2&1&3\end{bmatrix}$

In $\left[\begin{array}{cc}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{ll}6 & 2 \\ 5 & 8\end{array}\right]$ find the value of $x, y, z$ ?

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector $3\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}.$

Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by $\text{A}*\text{B}=\text{A}\cap\text{B}\ \ \forall\ \text{A},\text{ B}$ in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.