Calculus Problem Read Description

Question

Use the logarithmic differentiation process to PROVE the “scalar multiple” rule for derivatives.

Answer 1

Scalar Multiple Rule: If f(x) is differentiable, and c is a real number, then (cf(x))' = cf'(x).

Proof: [ln(cf(x))]' = [1/(cf(x))] (cf(x))' ( Recall that (lnu)' = u'/u)

But, [ln(cf(x))]' = [lnc + ln(f(x)]' = f'(x) / f(x) (Recall that ln(AB) = lnA + lnB)

So, [1/(cf(x))] (cf(x))' = f'(x) / f(x)

Therefore, (cf(x))' = cf(x)[f'(x)/f(x)] = cf'(x) QED