Derivatives of Power Functions

If $$f\left( x \right) = {x^p}$$, where $$p$$ is a real number, then

${\left( {{x^p}} \right)^\prime } = p{x^{p – 1}}.$

The derivation of this formula is given on the Definition of the derivative page.

If the exponent is a negative number, that is $$f\left( x \right) = {x^{ – p}}$$ $$\left( {p \gt0} \right),$$ then

${\left( {{x^{ – p}}} \right)^\prime } = { – p{x^{ – p – 1}} }={ – \frac{p}{{{x^{p + 1}}}}.}$

The Derivative of a Polynomial

Let $$f\left( x \right)$$ $$= {a_n}{x^n} + \ldots$$ $$+ {a_2}{x^2} + {a_1}x$$ $$+ {a_0}.$$ Then

${f’\left( x \right) }={ n{a_n}{x^{n – 1}} + \left( {n – 1} \right){a_{n – 1}}{x^{n – 2}} + \ldots} + {2{a_2}x + {a_1},}$

where $${a_n}$$, $${a_{n-1}}$$, $$\ldots$$, $${a_1}$$, $${a_0}$$, $$n$$ are constants. In particular, for a quadratic function:

${\left( {a{x^2} + bx + c} \right)^\prime } = {2ax + b,}$

where $$a$$, $$b$$, $$c$$ are constants.

The Derivative of an Irrational Function

If $$f\left( x \right) = \sqrt[\large m\normalsize]{x}$$, then such a function can be represented as a power function with exponent $$\large\frac{1}{m}\normalsize$$. Its derivative is given by

${f’\left( x \right) }= {\left( {\sqrt[\large m\normalsize]{x}} \right)^\prime } = {\frac{1}{{m\sqrt[\large m\normalsize]{{{x^{m – 1}}}}}}.}$

In particular, the derivative of the square root is

${f’\left( x \right) }= {\left( {\sqrt x } \right)^\prime } = {\frac{1}{{2\sqrt x }}.}$

Respectively, the derivative of the cubic root is

${f’\left( x \right) }= {\left( {\sqrt[\large 3\normalsize]{x}} \right)^\prime } = {\frac{1}{{3\sqrt[\large 3\normalsize]{{{x^2}}}}}.}$

Solved Problems

Click or tap a problem to see the solution.

Example 1

Calculate the derivative of the function $$y = 6{x^{100}} + 7{x^{50}} + 8x.$$

Example 2

Calculate the derivative of the function $$y = {\left( {\sqrt 3 } \right)^2} – 5\sqrt 2.$$

Example 3

Find the derivative of the function $$y = {\large\frac{1}{x}\normalsize} + {\large\frac{2}{{{x^2}}}\normalsize} + {\large\frac{3}{{{x^3}}}\normalsize}.$$

Example 4

Find the derivative of the following function: $$y = 8{x^5} – 6{x^4}$$ $$+ 5{x^3} – 7{x^2}$$ $$+ 4x + 3.$$

Example 5

Find the derivative of the function $$y = {\large\frac{{{x^2}}}{2}\normalsize} + {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{{x^4}}}{4}\normalsize}.$$

Example 6

Find the derivative of the function $$y = {\large\frac{{{x^2}}}{2}\normalsize} – {\large\frac{2}{{{x^2}}}\normalsize}.$$

Example 7

Differentiate $$y = {x^{10}} – \large{\frac{1}{{{x^{10}}}}}\normalsize.$$

Example 8

Calculate the value of the derivative of the function $$y = {x^2} – {\large\frac{1}{{2{x^2}}}\normalsize}$$ at $$x = 1.$$

Example 9

Find the derivative of the function $$y = \sqrt[\large 3\normalsize]{7}x + \sqrt[\large 7\normalsize]{3}.$$

Example 10

Find the derivative of the function $$y = \sqrt[\large 4\normalsize]{{{x^3}}}.$$

Example 11

Find the derivative of the function $$y = \sqrt[3]{{2{x^2}}}.$$

Example 12

Find the derivative of the irrational function $$y = \sqrt[\large m\normalsize]{{{x^n}}}$$ where $$m \ne 0.$$

Example 13

Calculate the derivative of the function $$y = \sqrt[\large\pi\normalsize]{{{x^2}}}.$$

Example 14

Find the derivative of the following function: $$y = x\left( {{x^2} + 2} \right)\left( {{x^3} – 3} \right).$$

Example 15

Calculate the derivative of the function $$y = \sqrt {\large\frac{x}{5}\normalsize} + \sqrt {\large\frac{5}{x}\normalsize}.$$

Example 16

Find the derivative of the function $$y = \sqrt[\large 3\normalsize]{x} – {\large\frac{1}{{\sqrt[3]{x}}}\normalsize}.$$

Example 17

Differentiate $$y = \large{\frac{1}{{\sqrt[4]{x}}}}\normalsize – \large{\frac{1}{{\sqrt[5]{x}}}}\normalsize.$$

Example 18

Find the derivative of the function $$y = 5{x^3} + 3 – {\large\frac{2}{{{x^3}}}\normalsize} + \sqrt[\large 3\normalsize]{{{x^5}}}.$$

Example 19

Find the derivative of the function $$y = {\large\frac{1}{x}\normalsize} + {\large\frac{1}{{\sqrt x }}\normalsize} + {\large\frac{1}{{\sqrt[3]{x}}}\normalsize}.$$

Example 20

Calculate the derivative of the function $$y = {\large\frac{2}{{\sqrt x }}\normalsize} + 3\sqrt[\large 3\normalsize]{x}.$$

Example 21

Find the derivative of the function $$y = \sqrt x – \sqrt[3]{x}.$$

Example 22

Find the derivative of the irrational function $$y = \sqrt {x\sqrt x }.$$

Example 23

Find the derivative of the function $${y = \sqrt {{x^2}\sqrt x }.}$$

Example 24

Find the derivative of the following irrational function: $$y = \sqrt[\large 3\normalsize]{{x\sqrt[\large 3\normalsize]{{{x^2}}}}}.$$

Example 25

Find the derivative of the function $$y = {\large\frac{3}{2}\normalsize} x\sqrt[\large 3\normalsize]{x}.$$

Example 26

Differentiate the function $$y = {\left( {1 – x} \right)^3}$$ without using the chain rule.

Example 1.

Calculate the derivative of the function $$y = 6{x^{100}} + 7{x^{50}} + 8x.$$

Solution.

First we apply the sum rule:

${y^\prime\left( x \right) }={ {\left( {6{x^{100}} + 7{x^{50}} + 8x} \right)^\prime } } = {{\left( {6{x^{100}}} \right)^\prime } + {\left( {7{x^{50}}} \right)^\prime } + {\left( {8x} \right)^\prime }.}$

By the constant multiple rule:

${y’\left( x \right) }= {6{\left( {{x^{100}}} \right)^\prime } + 7{\left( {{x^{50}}} \right)^\prime } + 8{\left( x \right)^\prime }.}$

Find the derivative of the power functions:

${y’\left( x \right) }={ 6 \cdot 100{x^{99}} + 7 \cdot 50{x^{49}} + 8 \cdot 1.}$

Simplifying and factoring, we have

${y’\left( x \right) = 600{x^{99}} + 350{x^{49}} + 8 } = {2\left( {300{x^{99}} + 175{x^{49}} + 4} \right).}$

Example 2.

Calculate the derivative of the function $$y = {\left( {\sqrt 3 } \right)^2} – 5\sqrt 2.$$

Solution.

The derivative of a constant is zero. Hence,

${y’\left( x \right) = {\left( {{{\left( {\sqrt 3 } \right)}^2} – 5\sqrt 2 } \right)^\prime } } = {{\left( {{{\left( {\sqrt 3 } \right)}^2}} \right)^\prime } – {\left( {5\sqrt 2 } \right)^\prime } }={ 0 – 0 = 0.}$

Example 3.

Find the derivative of the function $$y = {\large\frac{1}{x}\normalsize} + {\large\frac{2}{{{x^2}}}\normalsize} + {\large\frac{3}{{{x^3}}}\normalsize}.$$

Solution.

First we use the sum rule:

${y’\left( x \right) }={ {\left( {\frac{1}{x} + \frac{2}{{{x^2}}} + \frac{3}{{{x^3}}}} \right)^\prime } } = {{\left( {\frac{1}{x}} \right)^\prime } + {\left( {\frac{2}{{{x^2}}}} \right)^\prime } + {\left( {\frac{3}{{{x^3}}}} \right)^\prime }.}$

Then we apply the constant multiple rule and the power rule to get

${y’\left( x \right) }={ {\left( {\frac{1}{x}} \right)^\prime } + 2{\left( {\frac{1}{{{x^2}}}} \right)^\prime } + 3{\left( {\frac{1}{{{x^3}}}} \right)^\prime } } = {{\left( {{x^{ – 1}}} \right)^\prime } + 2{\left( {{x^{ – 2}}} \right)^\prime } + 3{\left( {{x^{ – 3}}} \right)^\prime } } = { – 1 \cdot {x^{ – 2}} + 2 \cdot \left( { – 2} \right){x^{ – 3}} }+{ 3 \cdot \left( { – 3} \right){x^{ – 4}} } = { – \frac{1}{{{x^2}}} – \frac{4}{{{x^3}}} – \frac{9}{{{x^4}}}.}$

Example 4.

Find the derivative of the following function: $$y = 8{x^5} – 6{x^4}$$ $$+ 5{x^3} – 7{x^2}$$ $$+ 4x + 3.$$

Solution.

Using the formula for differentiating a polynomial, we obtain the expression for the derivative in the form

${y’\left( x \right) }={ {\left( {8{x^5} – 6{x^4} + 5{x^3}} \right.}}\kern0pt{{-\left.{ 7{x^2} + 4x + 3} \right)^\prime } } = {{\left( {8{x^5}} \right)^\prime } – {\left( {6{x^4}} \right)^\prime } + {\left( {5{x^3}} \right)^\prime }} – {{\left( {7{x^2}} \right)^\prime } + {\left( {4x} \right)^\prime } + {\left( 3 \right)^\prime } } = {8 \cdot 5{x^4} – 6 \cdot 4{x^3} }+{ 5 \cdot 3{x^2} – 7 \cdot 2x }+{ 4 \cdot 1 + 0 } = {40{x^4} – 24{x^3} + 15{x^2} }-{ 14x + 4.}$

Example 5.

Find the derivative of the function $$y = {\large\frac{{{x^2}}}{2}\normalsize} + {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{{x^4}}}{4}\normalsize}.$$

Solution.

The derivative can be written as

${y’\left( x \right) }={ {\left( {\frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} + \frac{{{x^4}}}{4}} \right)^\prime } } = {{\left( {\frac{{{x^2}}}{2}} \right)^\prime } + {\left( {\frac{{{x^3}}}{3}} \right)^\prime } + {\left( {\frac{{{x^4}}}{4}} \right)^\prime } } = {\frac{1}{2}{\left( {{x^2}} \right)^\prime } + \frac{1}{3}{\left( {{x^3}} \right)^\prime } + \frac{1}{4}{\left( {{x^4}} \right)^\prime } } = {\frac{1}{2} \cdot 2x + \frac{1}{3} \cdot 3{x^2} + \frac{1}{4} \cdot 4{x^3} } = {x + {x^2} + {x^3} }={ x\left( {{x^2} + x + 1} \right).}$

Example 6.

Find the derivative of the function $$y = {\large\frac{{{x^2}}}{2}\normalsize} – {\large\frac{2}{{{x^2}}}\normalsize}.$$

Solution.

The derivative has the following form:

${y’\left( x \right) }={ {\left( {\frac{{{x^2}}}{2} – \frac{2}{{{x^2}}}} \right)^\prime } } = {{\left( {\frac{{{x^2}}}{2}} \right)^\prime } – {\left( {\frac{2}{{{x^2}}}} \right)^\prime } } = {\frac{1}{2}{\left( {{x^2}} \right)^\prime } – 2{\left( {\frac{1}{{{x^2}}}} \right)^\prime } } = {\frac{1}{2}{\left( {{x^2}} \right)^\prime } – 2{\left( {{x^{ – 2}}} \right)^\prime } } = {\frac{1}{2} \cdot 2x – 2 \cdot \left( { – 2} \right){x^{ – 3}} } = {x + 4{x^{ – 3}} }={ x + \frac{4}{{{x^3}}}.}$

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Problems 1-6
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Problems 7-26