# Find the derivative of (v/p)^0.5(bc*a)/(x^2)

## \frac{d}{dx}\left(\sqrt{\frac{v}{p}}\frac{ab\cdot c}{x^2}\right)

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$\frac{-2c\cdot b\cdot a\sqrt{v}}{x^{3}\sqrt{p}}$

## Step by step solution

Problem

$\frac{d}{dx}\left(\sqrt{\frac{v}{p}}\frac{ab\cdot c}{x^2}\right)$
1

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

$\sqrt{\frac{v}{p}}\cdot\frac{d}{dx}\left(\frac{c\cdot b\cdot a}{x^2}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\sqrt{\frac{v}{p}}\cdot\frac{x^2\frac{d}{dx}\left(c\cdot b\cdot a\right)-c\cdot b\cdot a\frac{d}{dx}\left(x^2\right)}{\left(x^2\right)^2}$
3

The derivative of the constant function is equal to zero

$\sqrt{\frac{v}{p}}\cdot\frac{0x^2-c\cdot b\cdot a\frac{d}{dx}\left(x^2\right)}{\left(x^2\right)^2}$
4

Any expression multiplied by $0$ is equal to $0$

$\sqrt{\frac{v}{p}}\cdot\frac{0-c\cdot b\cdot a\frac{d}{dx}\left(x^2\right)}{\left(x^2\right)^2}$
5

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

$\frac{0-1\cdot 2c\cdot b\cdot a\cdot x}{\left(x^2\right)^2}\sqrt{\frac{v}{p}}$
6

Multiply $2$ times $-1$

$\frac{0-2c\cdot b\cdot a\cdot x}{\left(x^2\right)^2}\sqrt{\frac{v}{p}}$
7

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

$\frac{-2c\cdot b\cdot a\cdot x}{\left(x^2\right)^2}\sqrt{\frac{v}{p}}$
8

Applying the power of a power property

$\frac{-2c\cdot b\cdot a\cdot x}{x^{4}}\sqrt{\frac{v}{p}}$
9

Simplifying the fraction by $x$

$\frac{-2c\cdot b\cdot a}{x^{3}}\sqrt{\frac{v}{p}}$
10

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

$\frac{-2c\cdot b\cdot a}{x^{3}}\cdot\frac{\sqrt{v}}{\sqrt{p}}$
11

Multiplying fractions

$\frac{-2c\cdot b\cdot a\sqrt{v}}{x^{3}\sqrt{p}}$

$\frac{-2c\cdot b\cdot a\sqrt{v}}{x^{3}\sqrt{p}}$

### Main topic:

Differential calculus

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