3.6 Derivatives of Logarithmic Functions 1Section 3.6 Derivatives of Log Functions

 In this section we use implicit differentiation to find the derivatives of the logarithmic functions y = log a x and, in particular, the natural logarithmic function y = ln x. [It can be proved that logarithmic functions are differentiable; this is certainly plausible from their graphs. Section 3.6 Derivatives of Log Functions2

Properties of logs. If a and b are positive numbers and n is rational, then the following properties are true: Section 3.6 Derivatives of Log Functions3

With Chain Rule Section 3.6 Derivatives of Log Functions4

Differentiate the functions. Section 3.6 Derivatives of Log Functions5

The calculation of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms. This method is called logarithmic differentiation. Section 3.6 Derivatives of Log Functions6

 Take the natural log of both sides of the equation.  Use the Laws of Logs to simplify.  Differentiate implicitly with respect to x.  Solve for dy/dx. Section 3.6 Derivatives of Log Functions7

Use log differentiation to find the derivative of the functions below. Section 3.6 Derivatives of Log Functions8

Differentiate the functions below. Section 3.6 Derivatives of Log Functions9

Find the first and second derivatives. Section 3.6 Derivatives of Log Functions10

Find an equation of the normal and tangent lines to the curve at the given point. Section 3.6 Derivatives of Log Functions11