# What is the derivative of f(x) = (x)(6^-2x)?

## I got 6^-2x - 2x(ln6). Is this correct?

Jan 29, 2018

${6}^{- 2 x} \left(1 - 2 x \ln 6\right)$

#### Explanation:

First, remember that:

$\frac{d}{\mathrm{dx}} \left({a}^{x}\right) = \frac{d}{\mathrm{dx}} \left({e}^{x \ln a}\right) = {e}^{x \ln a} \cdot \ln a = {a}^{x} \ln a$

So, using product rule, we can differentiate the expression as:

$\frac{d}{\mathrm{dx}} \left(x \cdot {6}^{- 2 x}\right) = \frac{d}{\mathrm{dx}} \left(x\right) \cdot {6}^{- 2 x} + x \cdot \frac{d}{\mathrm{dx}} {6}^{- 2 x}$

$= {6}^{- 2 x} + x \cdot \left({6}^{\textcolor{b l u e}{- 2 x}} \cdot \ln 6 \cdot \frac{d}{\mathrm{dx}} \left(\textcolor{b l u e}{- 2 x}\right)\right)$

$= {6}^{- 2 x} - 2 x \ln 6 \left({6}^{- 2 x}\right)$

$= {6}^{- 2 x} \left(1 - 2 x \ln 6\right)$

Your answer was very close to correct; you just forgot the ${6}^{- 2 x}$ in the second term. Hope this helps!