Chain Rule with a Logarithmic Function

Thus, applying the chain rule properly gives $y' = \frac{10*x}{5*x^2+1}$ for $y=\ln(5*x^2+1)$.

The derivative of $\ln(u)$ is $\frac{1}{u}$, so for $u=5*x^2+1$, one obtains $y' = \frac{1}{5*x^2+1}$ multiplied by the derivative of $u$.

One may differentiate $y=\ln(5*x^2+1)$ by taking the derivative of the inner function as $5*x$, leading to $y' = \frac{5*x}{5*x^2+1}$.

The correct derivative of the inner function $5*x^2+1$ is $10*x$.