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1

Here, we show you a step-by-step solved example of calculus. This solution was automatically generated by our smart calculator:

$\frac{d}{dx}\frac{\sqrt{x}}{\sin\left(x\right)}$
2

Apply the quotient rule for differentiation, 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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

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

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

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

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

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

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

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

Multiplying fractions $\frac{1}{2} \times \frac{1}{x^{\left|-\frac{1}{2}\right|}}$

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

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

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

Multiplying fractions $\frac{1}{2} \times \frac{1}{\sqrt{x}}$

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

Multiply $1$ times $1$

$\frac{1}{2\sqrt{x}}\sin\left(x\right)$
6

Multiplying fractions $\frac{1}{2} \times \frac{1}{\sqrt{x}}$

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

Multiply the fraction by the term

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

Any expression multiplied by $1$ is equal to itself

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

Multiply the fraction by the term

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

Combine all terms into a single fraction with $2\sqrt{x}$ as common denominator

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

Multiply $-1$ times $2$

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

When multiplying two powers that have the same base ($\sqrt{x}$), you can add the exponents

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

Cancel exponents $\frac{1}{2}$ and $2$

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

Divide fractions $\frac{\frac{\sin\left(x\right)-2x\cos\left(x\right)}{2\sqrt{x}}}{\sin\left(x\right)^2}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

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

Combine all terms into a single fraction with $2\sqrt{x}$ as common denominator

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

Endgültige Antwort auf das Problem

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

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