derivative of ln(x*sqrt(x^2-1)) — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} (\ln (x \cdot \sqrt{x^{2} - 1}))

We differentiate using:

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

(where f =

x \cdot \sqrt{x^{2} - 1}

)

ⓘ The derivative of ln(x) is 1/x:

\frac{d}{d x} (\ln (x \cdot \sqrt{x^{2} - 1}))

\frac{d}{d x} (x \cdot \sqrt{x^{2} - 1}) \cdot \frac{1}{x \cdot \sqrt{x^{2} - 1}}

\frac{d}{d x} (x \cdot \sqrt{x^{2} - 1}) \cdot \frac{1}{x \cdot \sqrt{x^{2} - 1}}

For

u1

x

and

v2

\sqrt{x^{2} - 1}

, use the differentiation rule:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's perform the differentiation

u1

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

ⓘ The derivative of x is 1:

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

1

1

3) Let's perform the differentiation

v2

\frac{d}{d x} (\sqrt{x^{2} - 1})

We differentiate using:

\frac{d}{d x} (\sqrt{f}) = \frac{d}{d x} (f) \cdot \frac{1}{2 \sqrt{f}}

(where f =

x^{2} - 1

)

ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

\frac{d}{d x} (\sqrt{x^{2} - 1})

\frac{d}{d x} (x^{2} - 1) \cdot \frac{1}{2 \cdot \sqrt{x^{2} - 1}}

\frac{d}{d x} (x^{2} - 1) \cdot \frac{1}{2 \cdot \sqrt{x^{2} - 1}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (x^{2} - 1) \cdot \frac{1}{2 \cdot \sqrt{x^{2} - 1}}

\frac{\frac{d}{d x} (x^{2} - 1) \cdot 1}{2 \cdot \sqrt{x^{2} - 1}}

\frac{\frac{d}{d x} (x^{2} - 1)}{2 \cdot \sqrt{x^{2} - 1}}

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

\frac{d}{d x} (x^{2} - 1)

\frac{d}{d x} (x^{2}) + 0

\frac{\frac{d}{d x} (x^{2}) + 0}{2 \cdot \sqrt{x^{2} - 1}}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (x^{2}) + 0

\frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

\frac{2 x}{2 \cdot \sqrt{x^{2} - 1}}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{2 x}{2 \cdot \sqrt{x^{2} - 1}}

\frac{x}{\sqrt{x^{2} - 1}}

\frac{x}{\sqrt{x^{2} - 1}}

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

1 \cdot \sqrt{x^{2} - 1} + x \cdot \frac{x}{\sqrt{x^{2} - 1}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

x \cdot \frac{x}{\sqrt{x^{2} - 1}}

\frac{x \cdot x}{\sqrt{x^{2} - 1}}

\frac{x^{2}}{\sqrt{x^{2} - 1}}

\sqrt{x^{2} - 1} + \frac{x^{2}}{\sqrt{x^{2} - 1}}

5) Then you can continue the differentiation:

ⅰ

\frac{d}{d x} (x \cdot \sqrt{x^{2} - 1})

\sqrt{x^{2} - 1} + \frac{x^{2}}{\sqrt{x^{2} - 1}}

\frac{\sqrt{x^{2} - 1} + \frac{x^{2}}{\sqrt{x^{2} - 1}}}{x \cdot \sqrt{x^{2} - 1}}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\sqrt{x^{2} - 1} + \frac{x^{2}}{\sqrt{x^{2} - 1}}

\frac{\sqrt{x^{2} - 1} \cdot \sqrt{x^{2} - 1} + x^{2}}{\sqrt{x^{2} - 1}}

\frac{\frac{\sqrt{x^{2} - 1} \cdot \sqrt{x^{2} - 1} + x^{2}}{\sqrt{x^{2} - 1}}}{x \cdot \sqrt{x^{2} - 1}}

ⓘ Multiplying powers with the same base adds the exponents:

\sqrt{x^{2} - 1} \cdot \sqrt{x^{2} - 1}

x^{2} - 1

\frac{\frac{x^{2} - 1 + x^{2}}{\sqrt{x^{2} - 1}}}{x \cdot \sqrt{x^{2} - 1}}

ⓘ Like terms are grouped by adding their coefficients:

x^{2} + x^{2}

2 x^{2}

\frac{\frac{2 x^{2} - 1}{\sqrt{x^{2} - 1}}}{x \cdot \sqrt{x^{2} - 1}}

ⓘ Dividing a fraction by

x \cdot \sqrt{x^{2} - 1}

is multiplying its denominator by

x \cdot \sqrt{x^{2} - 1}

\frac{\frac{2 x^{2} - 1}{\sqrt{x^{2} - 1}}}{x \cdot \sqrt{x^{2} - 1}}

\frac{2 x^{2} - 1}{\sqrt{x^{2} - 1} \cdot x \cdot \sqrt{x^{2} - 1}}

\frac{2 x^{2} - 1}{\sqrt{x^{2} - 1} \cdot x \cdot \sqrt{x^{2} - 1}}

ⓘ Multiplying powers with the same base adds the exponents:

\sqrt{x^{2} - 1} \cdot \sqrt{x^{2} - 1}

x^{2} - 1

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

▶Derivative:

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