Derive the function ln(xy^2-1y)+(3x-1y)^0.5 with respect to x

\frac{d}{dx}\left(\ln\left(x y^2-y\right)+\sqrt{3x-y}\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

$\frac{\frac{3}{2}}{\sqrt{3x-y}}+\frac{1}{xy^2-y}y^2$

Step by step solution

Problem

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

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

$\frac{d}{dx}\left(\sqrt{3x-y}\right)+\frac{d}{dx}\left(\ln\left(xy^2-y\right)\right)$
2

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

$\frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}\cdot\frac{d}{dx}\left(3x-y\right)+\frac{d}{dx}\left(\ln\left(xy^2-y\right)\right)$
3

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}\cdot\frac{d}{dx}\left(3x-y\right)+\frac{1}{xy^2-y}\cdot\frac{d}{dx}\left(xy^2-y\right)$
4

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

$\frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(-y\right)+\frac{d}{dx}\left(3x\right)\right)+\frac{1}{xy^2-y}\left(\frac{d}{dx}\left(-y\right)+\frac{d}{dx}\left(xy^2\right)\right)$
5

The derivative of the constant function is equal to zero

$\frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}\left(0+\frac{d}{dx}\left(3x\right)\right)+\frac{1}{xy^2-y}\left(0+\frac{d}{dx}\left(xy^2\right)\right)$
6

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

$\frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}\left(0+3\frac{d}{dx}\left(x\right)\right)+\frac{1}{xy^2-y}\left(0+y^2\frac{d}{dx}\left(x\right)\right)$
7

The derivative of the linear function is equal to $1$

$\left(0+1\cdot 3\right)\cdot \frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}+\frac{1}{xy^2-y}\left(0+y^2\frac{d}{dx}\left(x\right)\right)$
8

The derivative of the linear function is equal to $1$

$\left(0+1\cdot 3\right)\cdot \frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}+\left(0+1y^2\right)\frac{1}{xy^2-y}$
9

Multiply $3$ times $1$

$\left(0+3\right)\cdot \frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}+\left(0+1y^2\right)\frac{1}{xy^2-y}$
10

Add the values $3$ and $0$

$3\cdot \frac{1}{2}\left(3x-y\right)^{-\frac{1}{2}}+\left(0+1y^2\right)\frac{1}{xy^2-y}$
11

Multiply $\frac{1}{2}$ times $3$

$\frac{3}{2}\left(3x-y\right)^{-\frac{1}{2}}+\left(0+1y^2\right)\frac{1}{xy^2-y}$
12

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

$\frac{3}{2}\left(3x-y\right)^{-\frac{1}{2}}+1\frac{1}{xy^2-y}y^2$
13

Any expression multiplied by $1$ is equal to itself

$\frac{3}{2}\left(3x-y\right)^{-\frac{1}{2}}+\frac{1}{xy^2-y}y^2$
14

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

$\frac{3}{2}\cdot\frac{1}{\sqrt{3x-y}}+\frac{1}{xy^2-y}y^2$
15

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=\frac{3}{2}$ and $x=\sqrt{3x-y}$

$\frac{\frac{3}{2}}{\sqrt{3x-y}}+\frac{1}{xy^2-y}y^2$

Answer

$\frac{\frac{3}{2}}{\sqrt{3x-y}}+\frac{1}{xy^2-y}y^2$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.33 seconds

Views:

106