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 xnx^n (in other words, expressions with xx raised to any power):
ddxxn=nxn1\dfrac{d}{dx}x^n=n\cdot x^{n-1}
Basically, you take the power and multiply it by the expression, then you reduce the power by 11.
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 3x73x^7. We can differentiate it as follows:
ddx[3x7]=3ddx(x7)Constant multiple rule=3(7x6)Power rule=21x6\begin{aligned} \dfrac{d}{dx}[3x^7]&=3\dfrac{d}{dx}(x^7)\quad\gray{\text{Constant multiple rule}} \\\\ &=3(7x^6)\quad\gray{\text{Power rule}} \\\\ &=21x^6 \end{aligned}
Problem 1
f(x)=x5+2x3x2f(x)=x^5+2x^3-x^2
f(x)=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\dfrac{1}{x^2}, which is basically xx raised to a negative power. Consider this differentiation of 1x2\dfrac{1}{x^2}:
ddx(1x2)=ddx(x2)Rewrite as power=2x3Power rule=2x3Rewrite as fraction\begin{aligned} \dfrac{d}{dx}\left(\dfrac{1}{x^2}\right)&=\dfrac{d}{dx}(x^{-2})\quad\gray{\text{Rewrite as power}} \\\\ &=-2\cdot x^{-3}\quad\gray{\text{Power rule}} \\\\ &=-\dfrac{2}{x^3}\quad\gray{\text{Rewrite as fraction}} \end{aligned}
Problem 1
ddx(2x4+1x3x)=\dfrac{d}{dx}\left(\dfrac{-2}{x^4}+\dfrac{1}{x^3}-x\right)=

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\sqrt x or x23x^{^{\frac{2}{3}}}. Consider this differentiation of x\sqrt x:
ddxx=ddx(x12)Rewrite as power=12x12Power rule=12xRewrite as radical\begin{aligned} \dfrac{d}{dx}\sqrt x&=\dfrac{d}{dx}\left(x^{^{\Large\frac{1}{2}}}\right)\quad\gray{\text{Rewrite as power}} \\\\ &=\dfrac{1}{2}\cdot x^{^{\Large-\frac{1}{2}}}\quad\gray{\text{Power rule}} \\\\ &=\dfrac{1}{2\sqrt x}\quad\gray{\text{Rewrite as radical}} \end{aligned}
Problem 1
f(x)=6x23f(x)=6x^{^{\Large\frac{2}{3}}}
f(x)=f'(x)=

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