Question

What is the Power Rule?

Answer 1

If we have `y=x^n`, then we have a power rule for both differentiating and integrating

Differentiation

We "multiply by the power and subtract one from the power" :

Thus;

`dy/dx = d/dx (x^n) = nx^(n-1)`

Integration

We "add one to the power, and divide by that new power"

Thus:

`int \ y \ dx = int \ x^n \ dx = x^(n+1)/(n+1) + c`

Examples:

`d/dx( x^2 ) = 2x^(2-1) = 2x`
`d/dx( 3x^4 ) = 3(4)x^(4-1) = 12 x^3`

`int \ x^2 \ dx = x^(2+1)/(2+1) + c = x^3/3 + c`
`int \ 3x^4 \ dx = 3x^(4+1)/(4+1) + c = 3/5x^5 + c`