Calculus

Differentiation of Functions

Differentiation of Functions Logo

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 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:

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

    Page 1
    Problems 1-6
    Page 2
    Problems 7-26