Quick HELP

[jeflow]

How can I find f' of 5^3x.

 

[soroban]

Hello, jeflow!

How can I find f' of 5^3x.

You're expected to be familiar with the formula . . .

If \(\displaystyle y = b^u\), then: \(\displaystyle \,y'\:=\:b^u\cdot u'\cdot\ln(b)\)

 

[Unco]

How can I find f' of 5^3x.

Let \(\displaystyle \mbox{y = 5^{(3x)}}\)

Take natural logs of both sides and simplify

\(\displaystyle \mbox{ \ln{(y)} = 3x\cdot\ln{(5)}}\)

Now differentiate both sides implicitly wrt x

\(\displaystyle \mbox{ \frac{1}{y} \frac{dy}{dx} = 3\cdot\ln{(5)}}\)
\(\displaystyle \mbox{ }\)(\(\displaystyle \mbox{3\cdot\ln{(5)}}\) is just a constant)

Multiply both sides by y and remember that \(\displaystyle \mbox{y = 5^{3x}}\) so we have

\(\displaystyle \mbox{ \frac{dy}{dx} = 3\cdot\ln{(5)}\cdot 5^{(3x)}}\)