Find the derivative of (1-1cos(x))^0.5

\frac{d}{dx}\left(\sqrt{1-\cos\left(x\right)}\right)

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Answer

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

Step by step solution

Problem

$\frac{d}{dx}\left(\sqrt{1-\cos\left(x\right)}\right)$
1

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

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

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

$\frac{1}{2}\left(\frac{d}{dx}\left(-\cos\left(x\right)\right)+\frac{d}{dx}\left(1\right)\right)\left(1-\cos\left(x\right)\right)^{-\frac{1}{2}}$
3

The derivative of the constant function is equal to zero

$\frac{1}{2}\left(\frac{d}{dx}\left(-\cos\left(x\right)\right)+0\right)\left(1-\cos\left(x\right)\right)^{-\frac{1}{2}}$
4

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

$\frac{1}{2}\left(0-\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\left(1-\cos\left(x\right)\right)^{-\frac{1}{2}}$
5

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 $f'(x) = -\sin(x)\cdot D_x(x)$

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

Multiply $-1$ times $-1$

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

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

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

Multiply $1$ times $\frac{1}{2}$

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

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

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

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=\frac{1}{2}$ and $x=\sqrt{1-\cos\left(x\right)}$

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

Multiplying the fraction and term

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

Answer

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

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.26 seconds

Views:

136