Derivatives of Power Functions
Solved Problems
Example 11.
Find the derivative of the function \(y = \sqrt[3]{{2{x^2}}}.\)
Solution.
We rewrite the function as follows:
\[y = \sqrt[3]{{2{x^2}}} = \sqrt[3]{2} \cdot \sqrt[3]{{{x^2}}} = \sqrt[3]{2}{x^{\frac{2}{3}}}.\]
Using the constant multiple rule and the power rule, we have
\[y^\prime = \left( {\sqrt[3]{2}{x^{\frac{2}{3}}}} \right)^\prime = \sqrt[3]{2}\left( {{x^{\frac{2}{3}}}} \right)^\prime = \sqrt[3]{2} \cdot \frac{2}{3}{x^{\frac{2}{3} - 1}} = \sqrt[3]{2} \cdot \frac{2}{3}{x^{ - \frac{1}{3}}} = \frac{2}{3} \cdot {\left( {\frac{2}{x}} \right)^{\frac{1}{3}}} = \frac{2}{3}\sqrt[3]{{\frac{2}{x}}}.\]
Example 12.
Find the derivative of the irrational function \(y = \sqrt[m]{{{x^n}}}\) where \(m \ne 0.\)
Solution.
Differentiating as a power function with a fractional exponent, we have
\[y'\left( x \right) = \left( {\sqrt[m]{{{x^n}}}} \right)^\prime = \left( {{x^{\frac{n}{m}}}} \right)^\prime = \frac{n}{m}{x^{\frac{{n - m}}{m}}} = \frac{n}{m}{x^{ - \frac{{m - n}}{m}}} = \frac{n}{{m{x^{\frac{{m - n}}{m}}}}} = \frac{n}{{m\sqrt[m]{{{x^{m - n}}}}}}.\]
Example 13.
Calculate the derivative of the function \(y = \sqrt[\pi]{{{x^2}}}.\)
Solution.
The derivative of this power function is given by
\[y'\left( x \right) = \left( {\sqrt[\pi]{{{x^2}}}} \right)^\prime = \left( {{x^{\frac{2}{\pi }}}} \right)^\prime = \frac{2}{\pi }{x^{\frac{2}{\pi } - 1}} = \frac{2}{\pi }{x^{\frac{{2 - \pi }}{\pi }}} = \frac{2}{\pi }{x^{ - \frac{{\pi - 2}}{\pi }}} = \frac{2}{{\pi \sqrt[\pi]{{{x^{\pi - 2}}}}}}.\]
Example 14.
Find the derivative of the following function: \(y = x\left( {{x^2} + 2} \right)\left( {{x^3} - 3} \right).\)
Solution.
This function can be represented as a polynomial:
\[y = x\left( {{x^2} + 2} \right)\left( {{x^3} - 3} \right) = \left( {{x^3} + 2x} \right)\left( {{x^3} - 3} \right) = {x^6} + 2{x^4} - 3{x^3} - 6x.\]
Differentiating term by term, we obtain:
\[y'\left( x \right) = \left( {{x^6} + 2{x^4} - 3{x^3} - 6x} \right)^\prime = \left( {{x^6}} \right)^\prime + \left( {2{x^4}} \right)^\prime - \left( {3{x^3}} \right)^\prime - \left( {6x} \right)^\prime = 6{x^5} + 2 \cdot 4{x^3} - 3 \cdot 3{x^2} - 6 = 6{x^5} + 8{x^3} - 9{x^2} - 6.\]
Example 15.
Calculate the derivative of the function \(y = \sqrt {\frac{x}{5}} + \sqrt {\frac{5}{x}}.\)
Solution.
First, we rewrite the function as follows:
\[y\left( x \right) = \sqrt {\frac{x}{5}} + \sqrt {\frac{5}{x}} = \frac{1}{{\sqrt 5 }} \cdot \sqrt x + \sqrt 5 \cdot \frac{1}{{\sqrt x }}.\]
Use the sum rule for the derivative:
\[y'\left( x \right) = \left( {\frac{1}{{\sqrt 5 }} \cdot \sqrt x + \sqrt 5 \cdot \frac{1}{{\sqrt x }}} \right)^\prime = \left( {\frac{1}{{\sqrt 5 }} \cdot \sqrt x } \right)^\prime + \left( {\sqrt 5 \cdot \frac{1}{{\sqrt x }}} \right)^\prime .\]
Then we take out the constant factors and calculate the derivatives of the power functions:
\[y'\left( x \right) = \left( {\frac{1}{{\sqrt 5 }} \cdot \sqrt x } \right)^\prime + \left( {\sqrt 5 \cdot \frac{1}{{\sqrt x }}} \right)^\prime = \frac{1}{{\sqrt 5 }}{\left( {\sqrt x } \right)^\prime } + \sqrt 5 {\left( {\frac{1}{{\sqrt x }}} \right)^\prime } = \frac{1}{{\sqrt 5 }}{\left( {\sqrt x } \right)^\prime } + \sqrt 5 {\left( {{x^{ - \frac{1}{2}}}} \right)^\prime } = \frac{1}{{\sqrt 5 }} \cdot \frac{1}{{2\sqrt x }} + \sqrt 5 \cdot \left( { - \frac{1}{2}} \right){x^{ - \frac{1}{2} - 1}} = \frac{1}{{2\sqrt 5 \sqrt x }} - \frac{{\sqrt 5 }}{2}{x^{ - \frac{3}{2}}}. \]
Here we used the expression \({\left( {\sqrt x } \right)^\prime } = {\left( {{x^{\frac{1}{2}}}} \right)^\prime } =\) \({\frac{1}{2}}{x^{ - {\frac{1}{2}}}} = {\frac{1}{{2\sqrt x }}}.\) Simplifying, we have
\[y'\left( x \right) = \frac{1}{{2\sqrt 5 \sqrt x }} - \frac{{\sqrt 5 }}{2}{x^{ - \frac{3}{2}}} = \frac{1}{{2\sqrt 5 \sqrt x }} - \frac{{\sqrt 5 }}{{2{x^{\frac{3}{2}}}}} = \frac{{1 \cdot x}}{{2\sqrt 5 \sqrt x \cdot x}} - \frac{{\sqrt 5 \cdot \sqrt 5 }}{{2{x^{\frac{3}{2}}} \cdot \sqrt 5 }} = \frac{{x - 5}}{{2\sqrt 5 {x^{\frac{3}{2}}}}} = \frac{{x - 5}}{{2\sqrt {5{x^3}} }}.\]
Example 16.
Find the derivative of the function \(y = \sqrt[3]{x} - {\frac{1}{{\sqrt[3]{x}}}}.\)
Solution.
We turn to the expression in the power form:
\[y = \sqrt[3]{x} - \frac{1}{{\sqrt[3]{x}}} = {x^{\frac{1}{3}}} - {x^{ - \frac{1}{3}}}.\]
The derivative of the difference of two functions is equal to the difference of the derivatives of these functions:
\[y'\left( x \right) = \left( {{x^{\frac{1}{3}} - x^{ - \frac{1}{3}}}} \right)^\prime = \left( {{x^{\frac{1}{3}}}} \right)^\prime - \left( {{x^{ - \frac{1}{3}}}} \right)^\prime .\]
Calculating the derivatives of the power functions, we obtain:
\[y'\left( x \right) = \frac{1}{3}{x^{\frac{1}{3} - 1}} - \left( { - \frac{1}{3}} \right){x^{ - \frac{1}{3} - 1}} = \frac{1}{3}{x^{ - \frac{2}{3}}} + \frac{1}{3}{x^{ - \frac{4}{3}}} = \frac{1}{3}\left( {{x^{ - \frac{2}{3}} + x^{ - \frac{4}{3}}}} \right) = \frac{1}{3}\left( {\frac{1}{{{x^{\frac{2}{3}}}}} + \frac{1}{{{x^{\frac{4}{3}}}}}} \right) = \frac{1}{3}\left( {\frac{1}{{\sqrt[3]{{{x^2}}}}} + \frac{1}{{\sqrt[3]{{{x^4}}}}}} \right).\]
Example 17.
Differentiate \(y = \frac{1}{{\sqrt[4]{x}}} - \frac{1}{{\sqrt[5]{x}}}.\)
Solution.
First we convert the terms of the function to power form:
\[\frac{1}{{\sqrt[4]{x}}} = \frac{1}{{{x^{\frac{1}{4}}}}} = {x^{ - \frac{1}{4}}};\]
\[\frac{1}{{\sqrt[5]{x}}} = \frac{1}{{{x^{\frac{1}{5}}}}} = {x^{ - \frac{1}{5}}}.\]
Using the power rule, we obtain:
\[y^\prime = \left( {{x^{ - \frac{1}{4}}} - {x^{ - \frac{1}{5}}}} \right)^\prime = \left( {{x^{ - \frac{1}{4}}}} \right)^\prime - \left( {{x^{ - \frac{1}{5}}}} \right)^\prime = - \frac{1}{4}{x^{ - \frac{1}{4} - 1}} - \left( { - \frac{1}{5}} \right){x^{ - \frac{1}{5} - 1}} = - \frac{1}{4}{x^{ - \frac{5}{4}}} + \frac{1}{5}{x^{ - \frac{6}{5}}} = \frac{1}{{5{x^{\frac{6}{5}}}}} - \frac{1}{{4{x^{\frac{5}{4}}}}} = \frac{1}{{5\sqrt[5]{{{x^6}}}}} - \frac{1}{{4\sqrt[4]{{{x^5}}}}}.\]
Example 18.
Find the derivative of the function \(y = 5{x^3} + 3 - {\frac{2}{{{x^3}}}} + \sqrt[3]{{{x^5}}}.\)
Solution.
We convert each term of the function into a power form:
\[y = 5{x^3} + 3 - 2{x^{ - 3}} + {x^{\frac{5}{3}}}.\]
Using the linear properties of the derivative and the power rule, we have
\[y'\left( x \right) = \left( {5{x^3} + 3 - 2{x^{ - 3} + x^{\frac{5}{3}}}} \right)^\prime = \left( {5{x^3}} \right)^\prime + 3' - \left( {2{x^{ - 3}}} \right)^\prime + \left( {{x^{\frac{5}{3}}}} \right)^\prime = 5 \cdot 3{x^2} + 0 - 2 \cdot \left( { - 3} \right){x^{ - 3 - 1}} + \frac{5}{3}{x^{\frac{5}{3} - 1}} = 15{x^2} + 6{x^{ - 4}} + \frac{5}{3}{x^{\frac{2}{3}}} = 15{x^2} + \frac{6}{{{x^4}}} + \frac{{5\sqrt[3]{{{x^2}}}}}{3}.\]
Example 19.
Find the derivative of the function \(y = {\frac{1}{x}} + {\frac{1}{{\sqrt x }}} + {\frac{1}{{\sqrt[3]{x}}}}.\)
Solution.
Representing the terms in the form of power functions, we obtain the following expression for the derivative:
\[y'\left( x \right) = \left( {\frac{1}{x} + \frac{1}{{\sqrt x }} + \frac{1}{{\sqrt[3]{x}}}} \right)^\prime = \left( {\frac{1}{x}} \right)^\prime + \left( {\frac{1}{{\sqrt x }}} \right)^\prime + \left( {\frac{1}{{\sqrt[3]{x}}}} \right)^\prime = - \frac{1}{{{x^2}}} - \frac{1}{2}{x^{ - \frac{1}{2} - 1}} - \frac{1}{3}{x^{ - \frac{1}{3} - 1}} = - \frac{1}{{{x^2}}} - \frac{1}{2}{x^{ - \frac{3}{2}}} - \frac{1}{3}{x^{ - \frac{4}{3}}} = - \frac{1}{{{x^2}}} - \frac{1}{{2\sqrt {{x^3}} }} - \frac{1}{{3\sqrt[3]{{{x^4}}}}}.\]
Example 20.
Calculate the derivative of the function \(y = {\frac{2}{{\sqrt x }}} + 3\sqrt[3]{x}.\)
Solution.
Using the power rule, we get
\[y'\left( x \right) = \left( {\frac{2}{{\sqrt x }} + 3\sqrt[3]{x}} \right)^\prime = \left( {2{x^{ - \frac{1}{2}}} + 3{x^{\frac{1}{3}}}} \right)^\prime = \left( {2{x^{ - \frac{1}{2}}}} \right)^\prime + \left( {3{x^{\frac{1}{3}}}} \right)^\prime = 2\left( {{x^{ - \frac{1}{2}}}} \right)^\prime + 3\left( {{x^{\frac{1}{3}}}} \right)^\prime = 2 \cdot \left( { - \frac{1}{2}} \right){x^{ - \frac{1}{2} - 1}} + 3 \cdot \frac{1}{3}{x^{\frac{1}{3} - 1}} = - x^{ - \frac{3}{2}} + x^{ - \frac{2}{3}} = \frac{1}{{\sqrt[3]{{{x^2}}}}} - \frac{1}{{\sqrt {{x^3}} }}.\]
Example 21.
Find the derivative of the function \(y = \sqrt x - \sqrt[3]{x}.\)
Solution.
We convert the radical expressions to power form:
\[\sqrt x = {x^{\frac{1}{2}}},\;\;\sqrt[3]{x} = {x^{\frac{1}{3}}}.\]
Then applying the power rule, we get
\[y^\prime = \left( {{x^{\frac{1}{2}}} - {x^{\frac{1}{3}}}} \right)^\prime = \left( {{x^{\frac{1}{2}}}} \right)^\prime - \left( {{x^{\frac{1}{3}}}} \right)^\prime = \frac{1}{2}{x^{\frac{1}{2} - 1}} - \frac{1}{3}{x^{\frac{1}{3} - 1}} = \frac{1}{2}{x^{ - \frac{1}{2}}} - \frac{1}{3}{x^{ - \frac{2}{3}}} = \frac{1}{{2{x^{\frac{1}{2}}}}} - \frac{1}{{3{x^{\frac{2}{3}}}}} = \frac{1}{{2\sqrt x }} - \frac{1}{{3\sqrt[3]{{{x^2}}}}}.\]
Example 22.
Find the derivative of the irrational function \(y = \sqrt {x\sqrt x }.\)
Solution.
Converting the function to a power form, we obtain:
\[y'\left( x \right) = \left( {\sqrt {x\sqrt x } } \right)^\prime = \left( {\sqrt {x \cdot {x^{\frac{1}{2}}}} } \right)^\prime = \left( {\sqrt {{x^{\frac{3}{2}}}} } \right)^\prime = \left( {{{\left( {{x^{\frac{3}{2}}}} \right)}^{\frac{1}{2}}}} \right)^\prime = \left( {{x^{\frac{3}{4}}}} \right)^\prime = \frac{3}{4}{x^{\frac{3}{4} - 1}} = \frac{3}{4}{x^{ - \frac{1}{4}}} = \frac{3}{{4\sqrt[4]{x}}}.\]
Example 23.
Find the derivative of the function \({y = \sqrt {{x^2}\sqrt x }.}\)
Solution.
We convert the function to power form:
\[y\left( x \right) = \sqrt {{x^2}\sqrt x } = \sqrt {{x^2} \cdot {x^{\frac{1}{2}}}} = \sqrt {{x^{2 + \frac{1}{2}}}} = \sqrt {{x^{\frac{5}{2}}}} = {\left( {{x^{\frac{5}{2}}}} \right)^{\frac{1}{2}}} = {x^{\frac{5}{2} \cdot \frac{1}{2}}} = {x^{\frac{5}{4}}}.\]
Apply the power rule:
\[y^\prime\left( x \right) = \left( {{x^{\frac{5}{4}}}} \right)^\prime = \frac{5}{4}{x^{\frac{5}{4} - 1}} = \frac{5}{4}{x^{ - \frac{1}{4}}} = \frac{5}{{4{x^{\frac{1}{4}}}}} = \frac{5}{{4\sqrt[4]{x}}}.\]
Example 24.
Find the derivative of the following irrational function: \(y = \sqrt[3]{{x\sqrt[3]{{{x^2}}}}}.\)
Solution.
Similarly to the previous example, we have
\[y'\left( x \right) = \left( {\sqrt[3]{{x\sqrt[3]{{{x^2}}}}}} \right)^\prime = \left( {\sqrt[3]{{x \cdot {x^{\frac{2}{3}}}}}} \right)^\prime = \left( {\sqrt[3]{{{x^{\frac{5}{3}}}}}} \right)^\prime = \left( {{{\left( {{x^{\frac{5}{3}}}} \right)}^{\frac{1}{3}}}} \right)^\prime = \left( {{x^{\frac{5}{3} \cdot \frac{1}{3}}}} \right)^\prime = \left( {{x^{\frac{5}{9}}}} \right)^\prime = \frac{5}{9}{x^{\frac{5}{9} - 1}} = \frac{5}{9}{x^{ - \frac{4}{9}}} = \frac{5}{{9\sqrt[9]{{{x^4}}}}}.\]
Example 25.
Find the derivative of the function \(y = {\frac{3}{2}} x\sqrt[3]{x}.\)
Solution.
Differentiating this function as a power function, we obtain:
\[y'\left( x \right) = \left( {\frac{3}{2}x\sqrt[3]{x}} \right)^\prime = \left( {\frac{3}{2}x \cdot {x^{\frac{1}{3}}}} \right)^\prime = \frac{3}{2}{\left( {{x^{1 + \frac{1}{3}}}} \right)^\prime } = \frac{3}{2}{\left( {{x^{\frac{4}{3}}}} \right)^\prime } = \frac{3}{2} \cdot \frac{4}{3}{x^{\frac{4}{3} - 1}} = 2{x^{\frac{1}{3}}} = 2\sqrt[3]{x}.\]
Example 26.
Differentiate the function \(y = {\left( {1 - x} \right)^3}\) without using the chain rule.
Solution.
We apply the perfect cube identity
\[{\left( {a + b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}.\]
Hence
\[y = {\left( {1 - x} \right)^3} = 1 - 3x + 3{x^2} - {x^3}.\]
Then using the basic differentiation rules and the power rule, we find the derivative:
\[y^\prime = \left( {1 - 3x + 3{x^2} - {x^3}} \right)^\prime = 0 - 3 \cdot 1 + 3 \cdot 2x - 3{x^2} = - 3 + 6x - 3{x^2} = - 3\left( {1 - 2x + {x^2}} \right) = - 3{\left( {1 - x} \right)^2}.\]
