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 x, start superscript, n, end superscript (in other words, expressions with x raised to any power):
start fraction, d, divided by, d, x, end fraction, x, start superscript, n, end superscript, equals, n, dot, x, start superscript, n, minus, 1, end superscript
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 3, x, start superscript, 7, end superscript. 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, left parenthesis, x, right parenthesis, equals, x, start superscript, 5, end superscript, plus, 2, x, start superscript, 3, end superscript, minus, x, start superscript, 2, end superscript
f, prime, left parenthesis, x, right parenthesis, equals

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Differentiating negative powers

The Power rule also allows us to differentiate expressions like start fraction, 1, divided by, x, start superscript, 2, end superscript, end fraction, which is basically x raised to a negative power. Consider this differentiation of start fraction, 1, divided by, x, start superscript, 2, end superscript, end fraction:
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
start fraction, d, divided by, d, x, end fraction, left parenthesis, start fraction, minus, 2, divided by, x, start superscript, 4, end superscript, end fraction, plus, start fraction, 1, divided by, x, start superscript, 3, end superscript, end fraction, minus, x, right parenthesis, equals

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Differentiating fractional powers and radicals

The Power rule also allows us to differentiate expressions like square root of, x, end square root or x, start superscript, start superscript, start fraction, 2, divided by, 3, end fraction, end superscript, end superscript. Consider this differentiation of square root of, x, end square root:
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, left parenthesis, x, right parenthesis, equals, 6, x, start superscript, start superscript, start fraction, 2, divided by, 3, end fraction, end superscript, end superscript
f, prime, left parenthesis, x, right parenthesis, equals

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