Derivatives of Power Functions
If f (x) = xp, 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 (x) = x−p (p > 0), 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 \(f\left( x \right) = \sqrt[m]{x}\), then such a function can be represented as a power function with exponent \(\frac{1}{m}\). Its derivative is given by
\[f'\left( x \right) = \left( {\sqrt[m]{x}} \right)^\prime = \frac{1}{{m\sqrt[m]{{{x^{m - 1}}}}}}.\]
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
\[f'\left( x \right) = \left( {\sqrt[3]{x}} \right)^\prime = \frac{1}{{3\sqrt[3]{{{x^2}}}}}.\]
Solved Problems
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 \(y = {\frac{1}{x}} + {\frac{2}{{{x^2}}}} + {\frac{3}{{{x^3}}}}.\)
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
\[y'\left( x \right) = \left( {8{x^5} - 6{x^4} + 5{x^3} - 7{x^2} + 4x + 3} \right)^\prime = \left( {8{x^5}} \right)^\prime - \left( {6{x^4}} \right)^\prime + \left( {5{x^3}} \right)^\prime - \left( {7{x^2}} \right)^\prime + \left( {4x} \right)^\prime + \left( 3 \right)^\prime = 8 \cdot 5{x^4} - 6 \cdot 4{x^3} + 5 \cdot 3{x^2} - 7 \cdot 2x + 4 \cdot 1 + 0 = 40{x^4} - 24{x^3} + 15{x^2} - 14x + 4.\]
Example 5.
Find the derivative of the function \(y = {\frac{{{x^2}}}{2}} + {\frac{{{x^3}}}{3}} + {\frac{{{x^4}}}{4}}.\)
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 \(y = {\frac{{{x^2}}}{2}} - {\frac{2}{{{x^2}}}}.\)
Solution.
The derivative has the following form:
\[y'\left( x \right) = \left( {\frac{{{x^2}}}{2} - \frac{2}{{{x^2}}}} \right)^\prime = \left( {\frac{{{x^2}}}{2}} \right)^\prime - \left( {\frac{2}{{{x^2}}}} \right)^\prime = \frac{1}{2}{\left( {{x^2}} \right)^\prime } - 2 \left( {\frac{1}{{{x^2}}}} \right)^\prime = \frac{1}{2}{\left( {{x^2}} \right)^\prime } - 2 \left( {{x^{ - 2}}} \right)^\prime = \frac{1}{2} \cdot 2x - 2 \cdot \left( { - 2} \right){x^{ - 3}} = x + 4{x^{ - 3}} = x + \frac{4}{{{x^3}}}.\]
Example 7.
Differentiate \(y = {x^{10}} - \frac{1}{{{x^{10}}}}.\)
Solution.
Using the power rule, we get
\[y^\prime = \left( {{x^{10}} - \frac{1}{{{x^{10}}}}} \right)^\prime = \left( {{x^{10}}} \right)^\prime - \left( {\frac{1}{{{x^{10}}}}} \right)^\prime = \left( {{x^{10}}} \right)^\prime - \left( {{x^{ - 10}}} \right)^\prime = 10{x^{10 - 1}} - \left( { - 10{x^{ - 10 - 1}}} \right) = 10{x^9} + 10{x^{ - 11}} = 10{x^9} + \frac{{10}}{{{x^{11}}}}.\]
Example 8.
Calculate the value of the derivative of the function \(y = {x^2} - {\frac{1}{{2{x^2}}}}\) at \(x = 1.\)
Solution.
The derivative of this function has the form:
\[y'\left( x \right) = \left( {{x^2} - \frac{1}{{2{x^2}}}} \right)^\prime = \left( {{x^2}} \right)^\prime - \left( {\frac{1}{{2{x^2}}}} \right)^\prime = \left( {{x^2}} \right)^\prime - \frac{1}{2}{\left( {{x^{ - 2}}} \right)^\prime } = 2x - \frac{1}{2} \cdot \left( { - 2} \right){x^{ - 3}} = 2x + \frac{1}{{{x^3}}}.\]
The value of the derivative at \(x = 1\) is equal to
\[y'\left( 1 \right) = 2 \cdot 1 + \frac{1}{{{1^3}}} = 3.\]
Example 9.
Find the derivative of the function \(y = \sqrt[3]{7}x + \sqrt[7]{3}.\)
Solution.
Here, we are dealing with a linear function whose coefficients are irrational numbers. Hence,
\[y'\left( x \right) = \left( {\sqrt[3]{7}x + \sqrt[7]{3}} \right)^\prime = \left( {\sqrt[3]{7}x} \right)^\prime + \left( {\sqrt[7]{3}} \right)^\prime = \sqrt[3]{7} \cdot 1 + 0 = \sqrt[3]{7}.\]
Example 10.
Find the derivative of the function \(y = \sqrt[4]{{{x^3}}}.\)
Solution.
By representing this irrational function as a power function, we obtain:
\[y'\left( x \right) = \left( {\sqrt[4]{{{x^3}}}} \right)^\prime = \left( {{x^{\frac{3}{4}}}} \right)^\prime = \frac{3}{4}{x^{\frac{3}{4} - 1}} = \frac{3}{4}{x^{ - \frac{1}{4}}} = \frac{3}{{4\sqrt[4]{x}}}.\]
See more problems on Page 2.
