derivative of ln(y/x) — step-by-step solution

The expression

\ln (\frac{y}{x})

contains several variables: the variable x is chosen for derivation

1) Let's differentiate:

\frac{d}{d x} (\ln (\frac{y}{x}))

We differentiate using:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

(where f =

\frac{y}{x}

)

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (\frac{y}{x}))

\frac{d}{d x} (\frac{y}{x}) \cdot \frac{1}{\frac{y}{x}}

\frac{d}{d x} (\frac{y}{x}) \cdot \frac{1}{\frac{y}{x}}

For

p1

y

and

q2

x

, use the differentiation rule:

\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} (y)

ⓘ The derivative of a constant is zero:

\frac{d}{d x} (y)

0

0

3) Let's perform the differentiation

q2

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

ⓘ The derivative of x is 1:

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

1

1

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{0 \cdot x - y \cdot 1}{x^{2}}

ⓘ A product with a zero factor is zero:

0 \cdot x

0

\frac{0 - y}{x^{2}}

ⓘ Adding 0 changes nothing:

0 - y

- y

- \frac{y}{x^{2}}

5) Then you can continue the differentiation:

ⅰ

\frac{d}{d x} (\frac{y}{x})

- \frac{y}{x^{2}}

- \frac{\frac{y}{x^{2}}}{\frac{y}{x}}

ⓘ Dividing is multiplying by the reciprocal:

- \frac{\frac{y}{x^{2}}}{\frac{y}{x}}

- \frac{y}{x^{2}} \cdot \frac{x}{y}

- \frac{y}{x^{2}} \cdot \frac{x}{y}

We divide the numerator and denominator of

\frac{x}{x^{2}}

by the common factor:

x

, which gives the new fraction:

\frac{1}{x}

▶Derivative:

- \frac{1}{x}