6.3.2 Differentiating Logarithmic Functions

Question 1

Differentiating Logarithmic Functions

Answer 1

• Find the derivative of a general logarithmic function by rewriting it in terms of the natural logarithmic function and then differentiating.
• Differentiate composite functions involving logarithms by using the Chain Rule.
• Differentiate other complicated functions involving logarithms by using the standard rules of differentiation.

Question 2

note

Answer 2

• The derivative of a log function with a base different from e is not simply 1 over x. The base e has special properties that lead to that simple formula for the derivative of ln x. When the base is different from e, to find the derivative, the trick is to rewrite the function in terms of the natural log function.
• First, rewrite the log expression in exponential form. Then take the natural log of both sides. Finally, carry the exponent inside the log function outside to become a product. The result is an expression equivalent to the original function, but involving the natural log function (divided by a constant).
• Since the derivative of the natural log function is known, taking the derivative is now straightforward. In fact, the rule for the derivative of a general logarithmic function, y = log b x, is dy/dx = 1/[(ln b)x].
• When given a complicated function involving logarithms composed with other functions, the Chain Rule can be applied to find the derivative. If the logarithmic function has a base different from e, the rule above can be applied.
• In this example, first take the derivative of log 3 (blop), then take the derivative of the blop.
• There is sometimes more than one way to find the derivative. The blop in this example is sin 2 x, which is a function raised to a power. A property of logarithms says that the power inside the log can be changed to a product outside the log. Rewriting the function this way makes it possible to find the derivative with fewer steps.
• Other complicated functions might involve logarithms and products. The Product Rule can be applied just as it usually would.
• In this example, to find the derivative, use the Product Rule on the product (1 + x^2 )·ln(1 + x^2 ). The derivative of the natural log function requires the Chain Rule:
• d/dx (ln g(x)) = (1/g(x))g ́(x),
• where g(x) = 1 + x^2 .
• After applying the product rule, the answer can often be simplified greatly.

Question 3

Differentiate the given function.

f(x)=log10(sinxcosx)

Answer 3

f′(x)=cotx−tanx/ln10

Question 4

Differentiate the given function.

f(x)=ln(xcosx)

Answer 4

f′(x)=1−xtanx/x

Question 5

Differentiate the given function.

f(x)=log5(1/x^2)

Answer 5

f′(x)=−2/xln5

Question 6

Differentiate the given function.

f(x)=(1+2x)log2(3x)

Answer 6

f′(x)=2log2(3x)+1+2x/xln2

Question 7

Differentiate the given function.

f(x)=(x^2+2)ln(cos2x)

Answer 7

f′(x)=4xln(cosx)−(2x^2+4)tanx

Question 8

Differentiate the given function.

f(x)=xln(2x)

Answer 8

f′(x)=ln(2x)+1

Question 9

Differentiate the given function.

f(x)=log2(cos3x)

Answer 9

f′(x)=−3tanx/ln2

Question 10

Differentiate the given function.

f(x)=log5(1/sinx)

Answer 10

f′(x)=−cotx/ln5

Question 11

Differentiate the given function.

f(x)=sin^2xlog7(sin^3x)

Answer 11

f′(x)=6sinxcosxlog7(sinx) +3sinxcosx/ln7

Question 12

Differentiate the given function.

f(x)=x^2log3(x^3)

Answer 12

f′(x)=6xlog3x+3x/ln3