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

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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

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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

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With Chain Rule Section 3.6 Derivatives of Log Functions4

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Differentiate the functions. Section 3.6 Derivatives of Log Functions5

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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

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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

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

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Differentiate the functions below. Section 3.6 Derivatives of Log Functions9

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

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

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