Question

how would one integrate this function:

Answer 1

\[3^{(x(\ln(4x+5))/4}\]

Answer 2

and find the derivative...

Answer 3

I don't think you can integrate that

Answer 4

For differentiation, there's whole lot of steps, Rough guess
Power rule ---> Product rule ---> Chain rule ---> Product rule again ---> chain rule again

Answer 5

sorry, the integration was a typo - got that one mixed up with differentiation

Answer 6

\[\ \text{Use the chain rule}\]
\[\frac{d}{dx}\left(3^{\frac{1}{4} x \log (4 x+5)}\right)=\frac{d3^u}{du} \frac{du}{dx}\]
\[\text{where }u=\frac{1}{4} x \log (4 x+5)\text{ and }\frac{d3^u}{du}=3^u \log (3)\]
\[\log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{d}{dx}\left(\frac{1}{4} x \log (4 x+5)\right)\right)\]
\[\text{Use the product rule, }\frac{d}{dx}(u v)=v \frac{du}{dx}+u \frac{dv}{dx}\]
\[\text{where }u=x\text{ and }v=\log (4 x+5)\]
\[\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\log (4 x+5) \left(\frac{d}{dx}(x)\right)+x \left(\frac{d}{dx}(\log (4 x+5))\right)\right)\]
\[\text{Use the chain rule AGAIN, }\frac{d}{dx}(\log (4 x+5))=\frac{d\log (u)}{du} \frac{du}{dx}\]
\[\text{where }u=4 x+5\text{ and }\frac{d\log (u)}{du}=\frac{1}{u}\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(x \frac{\frac{d}{dx}(4 x+5)}{4 x+5}+\log (4 x+5) \left(\frac{d}{dx}(x)\right)\right)\]
\[\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{x \left(4 \left(\frac{d}{dx}(x)\right)+\frac{d}{dx}(5)\right)}{4 x+5}+\log (4 x+5) \left(\frac{d}{dx}(x)\right)\right)\]
Simplify the derivatives,
\[\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{4 x}{4 x+5}+\log (4 x+5)\right)\]
\[\log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{x}{4 x+5}+\frac{1}{4} \log (4 x+5)\right)\]