Find the derivative of 2^-3x with respect to x

Question

Question

Answer 1

Concept:

Formula:

log mn = n log m

\(\rm \frac{d(uv)}{dx} = v\frac{du}{dx}+u\frac{dv}{dx}\)

\(\rm \frac{d\log x}{dx} = \frac{1}{x}\)

Calculation:

Let y = 2^-3x

Taking log both sides, we get

⇒ log y = log 2^-3x

⇒ log y = -3x log 2 (∵ log mn = n log m)

⇒ log y = (-3log 2) x

differentiating with respect to x

⇒ \(\rm \frac 1 y \frac{dy}{dx} \) = (-3log 2) × 1

⇒ \(\rm \frac{dy}{dx} \) = (-3log 2)y

∴ \(\rm \frac{dy}{dx} \) = (-3log 2)2-3x