derivative of f(x)=(x+5)/(x-5) — step-by-step solution

We save the definition:

f (x) = \frac{x + 5}{x - 5}

1) Let's perform the differentiation:

\frac{d}{d x} (\frac{x + 5}{x - 5})

We differentiate

p1

x + 5

and

q2

x - 5

using:

\frac{d}{d x} (\frac{p1}{q2}) = \frac{\frac{d}{d x} (p1) \cdot q2 - p1 \cdot \frac{d}{d x} (q2)}{{q2}^{2}}

2) Let's differentiate

p1

\frac{d}{d x} (x + 5)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x + 5)

1 + 0

1 + 0

ⓘ Adding 0 changes nothing:

1 + 0

1

1

3) Let's perform the differentiation

q2

\frac{d}{d x} (x - 5)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x - 5)

1 + 0

1 + 0

ⓘ Adding 0 changes nothing:

1 + 0

1

1

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{1 \cdot (x - 5) - (x + 5) \cdot 1}{{(x - 5)}^{2}}

\frac{x - 5 - (x + 5)}{{(x - 5)}^{2}}

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (x + 5)

- x - 5

\frac{x - 5 - x - 5}{{(x - 5)}^{2}}

ⓘ The numerator and the denominator are computed before the division:

- 5 - 5

-10

■Like terms are grouped by adding their coefficients:

x - x

0

\frac{0 - 10}{{(x - 5)}^{2}}

ⓘ Adding 0 changes nothing:

0 - 10

-10

- \frac{10}{{(x - 5)}^{2}}

5) Then you can resume the differentiation:

ⅰ

\frac{d}{d x} (\frac{x + 5}{x - 5})

- \frac{10}{{(x - 5)}^{2}}

▶Derivative:

- \frac{10}{{(x - 5)}^{2}}