§ 4.5 The Derivative of ln x

Section Outline Derivatives for Natural Logarithms Differentiating Logarithmic Expressions

Derivative Rules for Natural Logarithms

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Differentiate. Use the power rule. Differentiate ln[g(x)]. Finish.

Differentiating Logarithmic Expressions EXAMPLE The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point? SOLUTION This is the given function. Use the quotient rule to differentiate. Simplify. Set the derivative equal to 0.

Differentiating Logarithmic Expressions CONTINUED The derivative will equal 0 when the numerator equals 0 and the denominator does not equal 0. Set the numerator equal to 0. Write in exponential form. To determine whether the function has a relative maximum at x = 1, let’s use the second derivative. This is the first derivative. Differentiate.

Differentiating Logarithmic Expressions CONTINUED Simplify. Factor and cancel. Evaluate the second derivative at x = 1. Since the value of the second derivative is negative at x = 1, the function is concave down at x = 1. Therefore, the function does indeed have a relative maximum at x = 1. To find the y-coordinate of this point So, the relative maximum occurs at (1, 1).