If you mean
![\dfrac{\mathrm d}{\mathrm dx}\left[4^x\ln(x)\right]](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B4%5Ex%5Cln%28x%29%5Cright%5D)
then first write 
. Then by the product rule,
![\dfrac{\mathrm d}{\mathrm dx}\left[4^x\ln(x)\right] = \ln(4)e^{\ln(4)x}\ln(x) + \dfrac{e^{\ln(4)x}}x = \ln(4)4^x\ln(x)+\dfrac{4^x}x=\dfrac{4^x}x\left(\ln(4)x\ln(x)+1\right)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B4%5Ex%5Cln%28x%29%5Cright%5D%20%3D%20%5Cln%284%29e%5E%7B%5Cln%284%29x%7D%5Cln%28x%29%20%2B%20%5Cdfrac%7Be%5E%7B%5Cln%284%29x%7D%7Dx%20%3D%20%5Cln%284%294%5Ex%5Cln%28x%29%2B%5Cdfrac%7B4%5Ex%7Dx%3D%5Cdfrac%7B4%5Ex%7Dx%5Cleft%28%5Cln%284%29x%5Cln%28x%29%2B1%5Cright%29)
If you instead mean
![\dfrac{\mathrm d}{\mathrm dx}\left[4^{x\ln(x)}\right]](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B4%5E%7Bx%5Cln%28x%29%7D%5Cright%5D)
you can again rewrite 
. Then by the chain and product rules,
![\dfrac{\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)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B4%5E%7Bx%5Cln%28x%29%7D%5Cright%5D%20%3D%20e%5E%7Bx%5Cln%28x%29%5Cln%284%29%7D%5Cln%284%29%5Cleft%28%5Cln%28x%29%2B1%5Cright%29%20%3D%204%5E%7Bx%5Cln%28x%29%7D%5Cln%284%29%28%5Cln%28x%29%2B1%29)
Step-by-step explanation:
For the first picture, we can see that as the multiplier of x gets larger (1/2 vs 1 vs 2), the line gets steeper
Next, we can see that as more stuff is added to x, the higher it goes. For example, y=x+2 is higher than y=x for all values of x
Lastly, we can see that multiplying x by a negative number can completely shift the positioning of the line, as seen by comparing y=-x to y=x
Answer the answer is f
Step-by-step explanation:
The answer is 6.25 pi cm squared
