Question

What is the value of \(\frac{d}{dx}(\frac{a^x}{e^x})\)?

(a) \(\frac{a^x (ln \,⁡a-a^x)}{e^x}\)

(b) \(\frac{a^x (ln\, ⁡a-e^x)}{e^x}\)

(c) \(\frac{a^x (ln\, ⁡a-1)}{e^x}\)

(d) \(\frac{a^x (ln\, ⁡a-1)}{(e^x)(e^x)}\)

The question was posed to me in class test.

Question is from Derivatives topic in section Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct choice is (c) \(\frac{a^x (ln\, ⁡a-1)}{e^x}\)

The best explanation: Using quotient rule, we know that, \(\frac{d}{dx} (\frac{f}{g}) = \frac{g.\frac{d}{dx} (f) – f.\frac{d}{dx}(g)}{g^2}\)

Here, f = a^x and g = e^x

\(\frac{d}{dx} (\frac{a^x}{e^x}) = \frac{e^x.\frac{d}{dx}(a^x)-a^x.\frac{d}{dx}(e^x)}{(e^x)(e^x)}\)   

\(\frac{d}{dx} (\frac{a^x}{e^x}) = \frac{e^x a^x ln⁡\, a-a^x e^x}{(e^x)(e^x)}\)

\(\frac{d}{dx} (\frac{a^x}{e^x}) = \frac{a^x (ln⁡ \,a-1)}{e^x}\)