Find the derivative of (v/p)^0.5(1-1(bc*a)/(x^2))

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

Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

e
π
ln
log
lim
d/dx
d/dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Answer

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

Step by step solution

Problem

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

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

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

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

$\sqrt{\frac{v}{p}}\left(\frac{d}{dx}\left(-\frac{c\cdot b\cdot a}{x^2}\right)+\frac{d}{dx}\left(1\right)\right)$
3

The derivative of the constant function is equal to zero

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

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

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

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

The derivative of the constant function is equal to zero

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

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

$\sqrt{\frac{v}{p}}\left(0-\frac{0-c\cdot b\cdot a\frac{d}{dx}\left(x^2\right)}{\left(x^2\right)^2}\right)$
8

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

$\left(0-\frac{0-1\cdot 2c\cdot b\cdot a\cdot x}{\left(x^2\right)^2}\right)\sqrt{\frac{v}{p}}$
9

Multiply $2$ times $-1$

$\left(0-\frac{0-2c\cdot b\cdot a\cdot x}{\left(x^2\right)^2}\right)\sqrt{\frac{v}{p}}$
10

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

$-\sqrt{\frac{v}{p}}\cdot\frac{-2c\cdot b\cdot a\cdot x}{\left(x^2\right)^2}$
11

Applying the power of a power property

$-\sqrt{\frac{v}{p}}\cdot\frac{-2c\cdot b\cdot a\cdot x}{x^{4}}$
12

Simplifying the fraction by $x$

$-\sqrt{\frac{v}{p}}\cdot\frac{-2c\cdot b\cdot a}{x^{3}}$
13

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

$-\frac{\sqrt{v}}{\sqrt{p}}\cdot\frac{-2c\cdot b\cdot a}{x^{3}}$
14

Multiplying fractions

$-\left(\frac{-2c\cdot b\cdot a\sqrt{v}}{\sqrt{p}x^{3}}\right)$
15

Multiplying the fraction and term

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

Answer

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

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.3 seconds

Views:

98