Find the derivative of (a+bx)/(abx*-1)

\frac{d}{dx}\left(\frac{a+bx}{a-b\cdot x}\right)

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Answer

$\frac{b\left(x\cdot b+a\right)+b\left(a-x\cdot b\right)}{\left(a-x\cdot b\right)^2}$

Step by step solution

Problem

$\frac{d}{dx}\left(\frac{a+bx}{a-b\cdot x}\right)$
1

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

$\frac{\left(a-x\cdot b\right)\frac{d}{dx}\left(x\cdot b+a\right)-\left(x\cdot b+a\right)\frac{d}{dx}\left(a-x\cdot b\right)}{\left(a-x\cdot b\right)^2}$
2

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

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

The derivative of the constant function is equal to zero

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

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

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

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

$\frac{\left(a-x\cdot b\right)\left(1b+0\right)-\left(x\cdot b+a\right)\left(\frac{d}{dx}\left(-x\cdot b\right)+0\right)}{\left(a-x\cdot b\right)^2}$
6

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=-b$ and $g=x$

$\frac{\left(a-x\cdot b\right)\left(1b+0\right)-\left(x\cdot b+a\right)\left(0-b\frac{d}{dx}\left(x\right)+x\frac{d}{dx}\left(-b\right)\right)}{\left(a-x\cdot b\right)^2}$
7

The derivative of the constant function is equal to zero

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

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

$\frac{\left(a-x\cdot b\right)\left(1b+0\right)-\left(x\cdot b+a\right)\left(0-b\frac{d}{dx}\left(x\right)+0\right)}{\left(a-x\cdot b\right)^2}$
9

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

$\frac{\left(a-x\cdot b\right)\left(1b+0\right)-\left(0+1\left(-1\right)b+0\right)\left(x\cdot b+a\right)}{\left(a-x\cdot b\right)^2}$
10

Add the values $0$ and $0$

$\frac{\left(a-x\cdot b\right)\left(1b+0\right)-1\cdot 1\left(-1\right)b\left(x\cdot b+a\right)}{\left(a-x\cdot b\right)^2}$
11

Multiply $-1$ times $-1$

$\frac{1b\left(x\cdot b+a\right)+\left(a-x\cdot b\right)\left(1b+0\right)}{\left(a-x\cdot b\right)^2}$
12

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

$\frac{1b\left(x\cdot b+a\right)+1b\left(a-x\cdot b\right)}{\left(a-x\cdot b\right)^2}$
13

Any expression multiplied by $1$ is equal to itself

$\frac{b\left(x\cdot b+a\right)+b\left(a-x\cdot b\right)}{\left(a-x\cdot b\right)^2}$

Answer

$\frac{b\left(x\cdot b+a\right)+b\left(a-x\cdot b\right)}{\left(a-x\cdot b\right)^2}$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.3 seconds

Views:

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