Calculus

Differentiation of Functions

Derivatives of Power Functions

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Problems 1-4
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Problems 5-20

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, i.e. \(f\left( x \right) = {x^{ – p}}\) \(\left( {p \gt0} \right),\) 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 on problem description to see 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

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

 Example 8

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

 Example 9

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

 Example 10

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

 Example 11

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

 Example 12

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

 Example 13

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

 Example 14

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

 Example 15

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

 Example 16

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 17

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

 Example 18

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

 Example 19

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

 Example 20

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

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

\[
{y’\left( x \right) }={ {\left( {8{x^5} – 6{x^4} + 5{x^3}} \right.}}\kern0pt{{-\left.{ 7{x^2} + 4x + 3} \right)^\prime } }
= {{\left( {8{x^5}} \right)^\prime } – {\left( {6{x^4}} \right)^\prime } + {\left( {5{x^3}} \right)^\prime }} – {{\left( {7{x^2}} \right)^\prime } + {\left( {4x} \right)^\prime } + {\left( 3 \right)^\prime } }
= {8 \cdot 5{x^4} – 6 \cdot 4{x^3} }+{ 5 \cdot 3{x^2} – 7 \cdot 2x }+{ 4 \cdot 1 + 0 }
= {40{x^4} – 24{x^3} + 15{x^2} }-{ 14x + 4.}
\]
Page 1
Problems 1-4
Page 2
Problems 5-20