Find the derivative of arcsin(sin(x)/4)

\frac{d}{dx}\left(arcsin\left(\sin\left(x\right)\cdot\frac{}{4}\right)\right)

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Answer

$\cos\left(x\right)\frac{}{4\sqrt{1-1\cdot \left(\frac{^2}{16}\right)\sin\left(x\right)^2}}$

Step by step solution

Problem

$\frac{d}{dx}\left(arcsin\left(\sin\left(x\right)\cdot\frac{}{4}\right)\right)$
1

Taking the derivative of arcsine

$\frac{d}{dx}\left(\frac{}{4}\sin\left(x\right)\right)\frac{1}{\sqrt{1-\left(\frac{}{4}\sin\left(x\right)\right)^2}}$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{}{4}\cdot\frac{1}{\sqrt{1-\left(\frac{}{4}\sin\left(x\right)\right)^2}}\cdot\frac{d}{dx}\left(\sin\left(x\right)\right)$
3

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{}{4}\cdot\frac{1}{\sqrt{1-\left(\frac{}{4}\sin\left(x\right)\right)^2}}\cos\left(x\right)$
4

The power of a product is equal to the product of it's factors raised to the same power

$\frac{}{4}\cdot\frac{1}{\sqrt{1-1\cdot \left(\frac{}{4}\right)^2\sin\left(x\right)^2}}\cos\left(x\right)$
5

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{}{4}\cdot\frac{1}{\sqrt{1-1\cdot \left(\frac{^2}{16}\right)\sin\left(x\right)^2}}\cos\left(x\right)$
6

Multiplying fractions

$\cos\left(x\right)\frac{}{4\sqrt{1-1\cdot \left(\frac{^2}{16}\right)\sin\left(x\right)^2}}$

Answer

$\cos\left(x\right)\frac{}{4\sqrt{1-1\cdot \left(\frac{^2}{16}\right)\sin\left(x\right)^2}}$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.34 seconds

Views:

91