Math CentralQuandaries & Queries


Question from Jesse, a student:

What is the derivative of (2^sinx)/(logbase4(2x+1))

Hi Jesse,

The first step is to apply the quotient rule but then you are faced with differentiating 2sin(x) and log4(2x + 1). I don't remember the differentiation expressions for af(x) and logb(f(x)), I just don't face these problems often enough to remember the expressions but I do know how to develop the expressions.

Differentiating g(x) = af(x).

Take the natural log of both sides to get

ln(g(x)) = ln(af(x)) = f(x) ln(a)

Differentiate both sides

g'(x)/g(x) = f '(x) ln(a)

Solve for g'(x)

g'(x) = g(x) f '(x) ln(a) = f '(x) af(x) ln(a)

Differentiating g(x) = logb(f(x)).

Rewrite the expression in exponential form.

bg(x) = f(x)

Take the natural log of each side

g(x) ln(b) = ln(f(x))

Differentiate both sides.

g'(x) ln(b) = f '(x)/f(x)

Solve for g'(x).

g'(x) = f '(x)/[f(x) ln(b)]

Use these expressions and the quotient rule to differentiate (2sinx)/(log4(2x+1))


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