Power rule - Derivatives

Power rule

When using the Definition of Derivative, finding the derivative of a long polynomial function with large exponents, or powers, can be very demanding. To avoid this, we introduce you one of the most powerful differentiation tools that simplifies this entire differentiation process – the Power Rule. In this section, we will see how the Power Rule allows us to easily derive the slope of a polynomial function at any given point.

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Notes:

POWER RULE: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$ , where $n$ is any real number

1.
power rule: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$
2.
constant multiple rule: $\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)$
4.
negative exponents: $\frac{1}{x} = {x^{ - 1}}$ and $\frac{1}{{{x^n}}} = {x^{ - n}}$
5.
rational exponents: $\sqrt x = {x^{\frac{1}{2}}}$ and ${^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}$

Power rule

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Lessons

Notes:

POWER RULE: ddx(xn)=nxn−1\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}​dx​​d​​(x​n​​)=nx​n−1​​ , where nnn is any real number

1.

power rule: ddx(xn)=nxn−1\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}​dx​​d​​(x​n​​)=nx​n−1​​

a)

ddx(x5){\;}\frac{{d}}{{{d}x}}\left( {{x^5}} \right)​dx​​d​​(x​5​​)

b)

ddx(x){\;}\frac{{d}}{{{d}x}}\left( x \right)​dx​​d​​(x)

c)

ddx(3){\;}\frac{{d}}{{{d}x}}\left( 3 \right)​dx​​d​​(3)

2.

constant multiple rule: ddx[cf(x)]=cddxf(x)\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)​dx​​d​​[cf(x)]=c​dx​​d​​f(x)

a)

ddx(4x3){\;}\frac{{d}}{{{d}x}}\left( {4{x^3}} \right)​dx​​d​​(4x​3​​)

b)

ddx(6x){\;}\frac{{d}}{{{d}x}}\left( {6x} \right)​dx​​d​​(6x)

c)

ddx(−x){\;}\frac{{d}}{{{d}x}}\left( { - x} \right)​dx​​d​​(−x)

3.

4.

negative exponents: 1x=x−1\frac{1}{x} = {x^{ - 1}}​x​​1​​=x​−1​​ and 1xn=x−n\frac{1}{{{x^n}}} = {x^{ - n}}​x​n​​​​1​​=x​−n​​

a)

ddx(1x2){\;}\frac{{d}}{{{d}x}}\left( {\frac{1}{{{x^2}}}} \right)​dx​​d​​(​x​2​​​​1​​)

b)

ddx(−53x){\;}\frac{{d}}{{{d}x}}\left( {\frac{{ - 5}}{{3x}}} \right)​dx​​d​​(​3x​​−5​​)

5.

rational exponents: x=x12\sqrt x = {x^{\frac{1}{2}}}√​x​​​=x​​2​​1​​​​ and bxa=xab{^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}​b​​√​x​a​​​​​=x​​b​​a​​​​

a)

ddx(3x5){\;}\frac{{d}}{{{d}x}}\left( {{^3}\sqrt{{{x^5}}}} \right)​dx​​d​​(​3​​√​x​5​​​​​)

b)

ddx(x){\;}\frac{{d}}{{{d}x}}\left( {\sqrt x } \right)​dx​​d​​(√​x​​​)

c)

ddx(821x3){\;}\frac{{d}}{{{d}x}}\left( {\frac{8}{{21\sqrt {{x^3}} }}} \right)​dx​​d​​(​21√​x​3​​​​​​​8​​)

Power rule

Don't just watch, practice makes perfect.

We have over 350 practice questions in Calculus for you to master.

or

POWER RULE: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$ , where $n$ is any real number

power rule: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$

${\;}\frac{{d}}{{{d}x}}\left( {{x^5}} \right)$

${\;}\frac{{d}}{{{d}x}}\left( x \right)$

${\;}\frac{{d}}{{{d}x}}\left( 3 \right)$

constant multiple rule: $\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)$

${\;}\frac{{d}}{{{d}x}}\left( {4{x^3}} \right)$

${\;}\frac{{d}}{{{d}x}}\left( {6x} \right)$

${\;}\frac{{d}}{{{d}x}}\left( { - x} \right)$

negative exponents: $\frac{1}{x} = {x^{ - 1}}$ and $\frac{1}{{{x^n}}} = {x^{ - n}}$

${\;}\frac{{d}}{{{d}x}}\left( {\frac{1}{{{x^2}}}} \right)$

${\;}\frac{{d}}{{{d}x}}\left( {\frac{{ - 5}}{{3x}}} \right)$

rational exponents: $\sqrt x = {x^{\frac{1}{2}}}$ and ${^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}$

${\;}\frac{{d}}{{{d}x}}\left( {{^3}\sqrt{{{x^5}}}} \right)$

${\;}\frac{{d}}{{{d}x}}\left( {\sqrt x } \right)$

${\;}\frac{{d}}{{{d}x}}\left( {\frac{8}{{21\sqrt {{x^3}} }}} \right)$