Hi Jesse,

The first step is to apply the quotient rule but then you are faced with differentiating 2^sin(x) and log₄(2x + 1). I don't remember the differentiation expressions for a^f(x) and log_b(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) = a^f(x).

Take the natural log of both sides to get

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

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

Rewrite the expression in exponential form.

b^g(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 (2^sinx)/(log₄(2x+1))

Harley