# What is the derivative of f(x)=sqrt(1+ln(x) ?

Sep 13, 2014

The derivative for this example involves the chain rule and the power rule. Convert the square root to an exponent. Then apply the Power Rule and the Chain Rule. Then simplify and remove the negative exponents.

$f \left(x\right) = \sqrt{1 + \ln \left(x\right)}$

$f \left(x\right) = {\left(1 + \ln \left(x\right)\right)}^{\frac{1}{2}}$

$f ' \left(x\right) = \left(\frac{1}{2}\right) {\left(1 + \ln \left(x\right)\right)}^{\left(\frac{1}{2}\right) - 1} \cdot \left(0 + \frac{1}{x}\right)$

$f ' \left(x\right) = \left(\frac{1}{2}\right) {\left(1 + \ln \left(x\right)\right)}^{\left(- \frac{1}{2}\right)} \cdot \left(\frac{1}{x}\right)$

$f ' \left(x\right) = \left(\frac{1}{2 x}\right) {\left(1 + \ln \left(x\right)\right)}^{\left(- \frac{1}{2}\right)}$

$f ' \left(x\right) = \frac{1}{2 x \sqrt{1 + \ln \left(x\right)}}$