Logarithmic Differentiation – Solved Example (3)

Since both exponent and base contain variables/algebraic expressions of the differentiated variable ‘x’, so this derivative cannot be completed by the power rule or exponential rule of derivatives. However using logarithmic differentiation it can be done as follows:
Take natural logarithm on both sides,
Apply logarithm laws to simplify expression,

Differentiate with respect to x:
As discussed in the previous examples, derivative of ln(f(x)) is f `(x)/f(x). Apply product rule on the right side:
Apply chain rule on the derivative of ln(x^2 + 1)
Cross multiply f(x) and substitute its original expression:

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