Derivatives of more complex functions

First differentiate the logarithm. Differentiate the reciprocal linear expression again, keeping the minus sign and the inner derivative. Then substitute x = 0.

For ln(u), differentiate u and divide by u. Differentiate the quadratic inside the logarithm, then substitute the specified x.

Differentiate the square, then the sine and its argument. The resulting product can be simplified with the double-angle identity. Pay attention to the sign of the specified point.

Use the quotient rule: numerator derivative times denominator minus numerator times denominator derivative, divided by the denominator squared. Then substitute the point.

Use the chain rule: divide the inner derivative by twice the square root of the inner expression. Check that the radicand is positive at the specified point.

Apply the product rule. Differentiating the exponential adds the derivative of its linear exponent as a factor. Add both product-rule terms and substitute x = 0.