Derivative of composite function including natural logarithm/ln.

**flashylightsmeow** · March 1st 2011, 10:29 AM

Hi, I'm having a problem reconciling my solution with the book I'm working from.
The question is:
Calculate the derivative of:

$\ln(x+\sqrt{x^2+1})$

using the chain rule I get as far as
$\frac{1}{x+\sqrt{x^2+1}}(1+\frac{x}{\sqrt{x^2+1}})$

From here I would multiply each term inside the bracket by the fraction outside and get two terms to add together. BUT the book I'm using states that the above equation:

$\frac{1}{x+\sqrt{x^2+1}}(1+\frac{x}{\sqrt{x^2+1}}) =\frac{1}{x+\sqrt{x^2+1}}\frac{x+\sqrt{x^2+1}}{\sq rt{x^2+1}}$

Giving a final answer of:
$\frac{1}{\sqrt{x^2+1}}$

But I can't for the life of me see how the author got that. Surely multiplying the two terms in the chain rule will give the sum of two fractions!?!??! If someone could tell me where I'm going wrong I would appreciate it.

May thanks.

**Quacky** · March 1st 2011, 10:40 AM

Notice that:
$\displaystyle 1+\frac{x}{\sqrt{x^2+1}}$

$=\displaystyle\frac{\sqrt{x^2+1}}{\sqrt{x^2+1}}+\f rac{x}{\sqrt{x^2+1}}$

$=\displaystyle\frac{x+\sqrt{x^2+1}}{\sqrt{x^2+1}}$
So they've rewritten the second term as a combined fraction, before carrying out the multiplication.

**flashylightsmeow** · March 1st 2011, 11:06 AM

Quacky, you're a monkey genius. And I should stop trying to do maths. Many thanks!

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