MHB Using the Product Rule to Solve $\d{}{x}{3}^{x}\ln\left({3}\right)$

[karush]

$\d{}{x}{3}^{x}\ln\left({3}\right)=$
I tried the product rule but didn't get the answer

 

[Dethrone]

Hi karush,

You do not need product rule. $\ln(3)$ is a constant.

 

[karush]

How about the $3^x$

 

[I like Serena]

How about the $3^x$

$$3^x = e^{x\ln 3}$$

 

[Prove It]

How about the $3^x$

$\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \, \left( a^x \right) = a^x\,\ln{(a)} \end{align*}$

Proof:

$\displaystyle \begin{align*} y &= a^x \\ \ln{(y)} &= \ln{ \left( a^x \right) } \\ \ln{(y)} &= x\ln{(a)} \\ \frac{\mathrm{d}}{\mathrm{d}x} \, \left[ \ln{(y)} \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \, \left[ x\ln{(a)} \right] \\ \frac{\mathrm{d}}{\mathrm{d}y} \, \left[ \ln{(y)} \right] \, \frac{\mathrm{d}}{\mathrm{d}x} \, \left( y \right) &= \ln{(a)} \\ \frac{1}{y}\,\frac{\mathrm{d}y}{\mathrm{d}x} &= \ln{(a)} \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= y\ln{(a)} \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= a^x\,\ln{(a)} \end{align*}$

Q.E.D.