Question

How do you differentiate `y=e^x+x^10-1/x`?

Answer 1

`color(blue)(y'(x) = e^x + 10x^9 + 1/(x^2)`

Explanation:

We're asked to find the derivative

`(dy)/(dx) [y = e^x + x^10 - 1/x]`

The derivative of `e^x` is defined as `e^x`:

`y'(x) = e^x + d/(dx)[x^10] - d/(dx)[1/x]`

Us the power rule on the `x^10` term:

`y'(x) = e^x + 10x^9 - d/(dx)[1/x]`

Use the quotient rule on the `1/x` term, which states

`d/(dx)[u/v] = (v(du)/(dx) - u(dv)/(dx))/(v^2)`

where

• `u = 1`

• `v = x`:

`y'(x) = e^x + 10x^9 - (xd/(dx)[1] - 1d/(dx)[x])/(x^2)`

The derivative of `1` is `0` and the derivative of `x` is `1`:

`color(blue)(y'(x) = e^x + 10x^9 + 1/(x^2)`