Power rule for differentiation
The power rule allows us to easy differentiate any function, as long as we
only have powers of
To differentiate a term
- Multiply the term by the power
: . - Decrease the power by 1:
. - So the derivative of
is: .
Differentiating larger expressions
Section titled “Differentiating larger expressions”Example: Differentiate
Section titled “Example: Differentiate ”- Differentiate each term separately:
: Multiply by 4 and decrease power by 1: . : Multiply by 3 and decrease power by 1: . : Multiply by 2 and decrease power by 1: . : Multiply by 1 and decrease power by 1: . : Constant term, derivative is .
- Add together the results:
.
- Answer:
.
Example: Differentiate
Section titled “Example: Differentiate ”- Differentiate each term separately:
: Multiply by 5 and decrease power by 1: . : Multiply by 4 and decrease power by 1: . : Multiply by 2 and decrease power by 1: . : Constant term, derivative is .
- Add together the results:
.
- Answer:
.
Example: Differentiate
Section titled “Example: Differentiate ”- Differentiate each term separately:
: Multiply by 3 and decrease power by 1: . : Multiply by 1 and decrease power by 1: . : Constant term, derivative is .
- Add together the results:
.
- Answer:
.
Example: Differentiate
Section titled “Example: Differentiate ”- Differentiate each term separately:
: Multiply by 6 and decrease power by 1: . : Multiply by 3 and decrease power by 1: . : Multiply by 2 and decrease power by 1: . : Multiply by 1 and decrease power by 1: . : Constant term, derivative is .
- Add together the results:
.
- Answer:
.
Differentiating terms with negative or fractional powers
Section titled “Differentiating terms with negative or fractional powers”We can use the exact same power rule to differentiate terms with negative or fractional powers.
Make sure to remember that, for negative powers, decreasing the power by 1 means
making it more negative (e.g. from
Example: Differentiate
Section titled “Example: Differentiate ”- Differentiate each term separately:
: Multiply by -3 and decrease power by 1: . : Multiply by 1/2 and decrease power by 1: . : Constant term, derivative is .
- Add together the results:
.
- Answer:
.
Example: Differentiate
Section titled “Example: Differentiate ”- Differentiate each term separately:
: Multiply by 3/2 and decrease power by 1: . : Multiply by -1 and decrease power by 1: . : Constant term, derivative is .
- Add together the results:
.
- Answer:
.