Question

Differentiate 4^xlnx

Answer 1

If you mean

`(\mathrm d)/(\mathrm dx)\left[4^x\ln(x)\right]`

then first write
`4^x=e^(\ln(4^x))=e^(\ln(4)x)`. Then by the product rule,

`(\mathrm d)/(\mathrm dx)\left[4^x\ln(x)\right] = \ln(4)e^(\ln(4)x)\ln(x) + \frac{e^(\ln(4)x)}x = \ln(4)4^x\ln(x)+\frac{4^x}x=\frac{4^x}x\left(\ln(4)x\ln(x)+1\right)`

If you instead mean

`(\mathrm d)/(\mathrm dx)\left[4^(x\ln(x))\right]`

you can again rewrite
`4^(x\ln(x)) = e^{\ln\left(4^(x\ln(x))\right)}=e^(x\ln(x)\ln(4))`. Then by the chain and product rules,

`(\mathrm d)/(\mathrm dx)\left[4^(x\ln(x))\right] = e^(x\ln(x)\ln(4))\ln(4)\left(\ln(x)+1\right) = 4^(x\ln(x))\ln(4)(\ln(x)+1)`