Find the derivative of ((cos(x)-1)/(sin(x)))^(1/2)

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

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Answer

$\frac{1}{2}\left(\frac{-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sqrt{\sin\left(x\right)^{3}}\sqrt{\cos\left(x\right)-1}}\right)$

Step by step solution

Problem

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

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{1}{2}\cdot\frac{\sin\left(x\right)\frac{d}{dx}\left(\cos\left(x\right)-1\right)-\frac{d}{dx}\left(\sin\left(x\right)\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\left(\frac{\cos\left(x\right)-1}{\sin\left(x\right)}\right)^{-\frac{1}{2}}$
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 ${f'(x) = \cos(x)\cdot D_x(x)}$

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

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{1}{2}\cdot\frac{\sin\left(x\right)\left(\frac{d}{dx}\left(-1\right)+\frac{d}{dx}\left(\cos\left(x\right)\right)\right)-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\left(\frac{\cos\left(x\right)-1}{\sin\left(x\right)}\right)^{-\frac{1}{2}}$
5

The derivative of the constant function is equal to zero

$\frac{1}{2}\cdot\frac{\sin\left(x\right)\left(0+\frac{d}{dx}\left(\cos\left(x\right)\right)\right)-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\left(\frac{\cos\left(x\right)-1}{\sin\left(x\right)}\right)^{-\frac{1}{2}}$
6

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

$\frac{1}{2}\cdot\frac{\sin\left(x\right)\left(0-\sin\left(x\right)\right)-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\left(\frac{\cos\left(x\right)-1}{\sin\left(x\right)}\right)^{-\frac{1}{2}}$
7

$x+0=x$, where $x$ is any expression

$\frac{1}{2}\cdot\frac{-\sin\left(x\right)\sin\left(x\right)-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\left(\frac{\cos\left(x\right)-1}{\sin\left(x\right)}\right)^{-\frac{1}{2}}$
8

When multiplying exponents with same base you can add the exponents

$\frac{1}{2}\cdot\frac{-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\left(\frac{\cos\left(x\right)-1}{\sin\left(x\right)}\right)^{-\frac{1}{2}}$
9

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

$\frac{1}{2}\cdot\frac{-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sin\left(x\right)^2}\cdot\frac{\left(\cos\left(x\right)-1\right)^{-\frac{1}{2}}}{\sin\left(x\right)^{-\frac{1}{2}}}$
10

Multiplying fractions

$\frac{1}{2}\left(\frac{\left(-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)\right)\left(\cos\left(x\right)-1\right)^{-\frac{1}{2}}}{\sqrt{\sin\left(x\right)^{3}}}\right)$
11

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1}{2}\left(\frac{\left(-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)\right)\frac{1}{\sqrt{\cos\left(x\right)-1}}}{\sqrt{\sin\left(x\right)^{3}}}\right)$
12

Multiplying the fraction and term

$\frac{1}{2}\left(\frac{\frac{-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sqrt{\cos\left(x\right)-1}}}{\sqrt{\sin\left(x\right)^{3}}}\right)$
13

Simplifying the fraction

$\frac{1}{2}\left(\frac{-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sqrt{\sin\left(x\right)^{3}}\sqrt{\cos\left(x\right)-1}}\right)$

Answer

$\frac{1}{2}\left(\frac{-\sin\left(x\right)^2-\cos\left(x\right)\left(\cos\left(x\right)-1\right)}{\sqrt{\sin\left(x\right)^{3}}\sqrt{\cos\left(x\right)-1}}\right)$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.4 seconds

Views:

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