Question

What is the derivative of `y=xln(x)+7x`?

Answer 1

First of all, let's observe that the derivative of a sum is the sum of the derivatives. So,
`d/dx (x\ln(x) + 7x) = d/dx x\ln(x) + d/dx 7x`

The easy part is calculating that `d/dx 7x = 7 d/dx x`: for the power rule `d/dx x^n=nx^{n-1}`, applied with `n=1`, one has that `d/dx x = 1`, and so `d/dx 7x=7`.

As for `x\ln(x)`, we apply the product rule, which says that `(fg)'=f'g+fg'`. In your case, `f(x)=x` and `g(x)=\ln(x)`. We have that `f'(x)=1` and `g'(x)=1/x`. So, `d/dx x\ln(x) = 1\cdot \ln(x) + x \cdot 1/x`

Finally, the answer is
`d/dx (x\ln(x) + 7x) = \ln(x)+1+7=\ln(x)+8`