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Power rule review

Review your knowledge of the Power rule for derivatives and solve problems with it.

What is the Power rule?

The Power rule tells us how to differentiate expressions of the form xn (in other words, expressions with x raised to any power):
ddxxn=nxn1
Basically, you take the power and multiply it by the expression, then you reduce the power by 1.
Want to learn more about the Power rule? Check out this video.

Differentiating polynomials

The Power rule, along with the more basic differentiation rules, allows us to differentiate any polynomial. Consider, for example, the monomial 3x7. We can differentiate it as follows:
ddx[3x7]=3ddx(x7)Constant multiple rule=3(7x6)Power rule=21x6
Problem 1
f(x)=x5+2x3x2
f(x)=

Want to try more problems like this? Check out this exercise.

Differentiating negative powers

The Power rule also allows us to differentiate expressions like 1x2, which is basically x raised to a negative power. Consider this differentiation of 1x2:
ddx(1x2)=ddx(x2)Rewrite as power=2x3Power rule=2x3Rewrite as fraction
Problem 1
ddx(2x4+1x3x)=

Want to try more problems like this? Check out this exercise.

Differentiating fractional powers and radicals

The Power rule also allows us to differentiate expressions like x or x23. Consider this differentiation of x:
ddxx=ddx(x12)Rewrite as power=12x12Power rule=12xRewrite as radical
Problem 1
f(x)=6x23
f(x)=

Want to try more problems like this? Check out these exercises:

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