Question

find derivative of x9^x

Answer 1

please explain steps too

Answer 2

Go! Looks like you have a product rule on your hands. Let's see your best work.

Answer 3

it says i should rewrite using logarithmic properties before differentiating

Answer 4

oh one sec let me give it a try

Answer 5

product rule: 9^x + xln9(9^x)

Answer 6

right?

Answer 7

I don't know why you would do that for this one, but okay.

y = x9^x
Then
log y = log(x9^x) = log(x) + log(9^x) = log(x) + xlog(9)

Now, you have an implicit definition, requiring the Chain Rule on the left. The right is pretty straight-forward.

Answer 8

1/yln(10)?

Answer 9

book has 9^x(xln9+1)

Answer 10

you got the book answer with the product rule. To pick up from tk:

\[y = x9^x\]
\[\ln y=\ln(x9^x)\]
\[\ln y = \ln x + \ln 9^x\]
\[\ln y = \ln x + x \ln 9\]
\[\frac{ 1 }{ y }\frac{ dy }{ dx }=\frac{1}{x}+ln 9\]
\[\frac{ dy }{ dx }=y \left( \frac{ 1 }{ x }+\ln 9 \right)\]
\[\frac{ dy }{ dx }=x9^x \left( \frac{ 1 }{ x }+\ln 9 \right)\]
\[\frac{ dy }{ dx }=9^x \left( 1+x \ln 9 \right)\]

I really don't see why they'd have you do it this way, unless it was just for practice.

Answer 11

That was my impression. Practice introducing the logarithm. Note to OP, You WILL need the logarithm for derivatives like \(\dfrac{d}{dx}x^{x}\) or limits like \(\underset{{x\to \infty}}{\lim}\left(1 + \dfrac{a}{x}\right)^{x}\). With the variable in both Base and Exponent, it takes just a bit more effort.