Calculus

Differentiation of Functions

Differentiation of Functions Logo

Basic Differentiation Rules

  • The operation of differentiation or finding the derivative of a function has the fundamental property of linearity. This property makes taking the derivative easier for functions constructed from the basic elementary functions using the operations of addition and multiplication by a constant number. The basic differentiation rules allow us to compute the derivatives of such functions without using the formal definition of the derivative. Consider these rules in more detail.

    Derivative of a Constant

    If \(f\left( x \right) = C,\) then

    \[f’\left( x \right) = C’ = 0.\]

    The proof of this rule is considered on the Definition of the Derivative page.

    Constant Multiple Rule

    Let \(k\) be a constant. If \(f\left( x \right)\) is differentiable, then \(kf\left( x \right)\) is also differentiable and

    \[{\left( {kf\left( x \right)} \right)^\prime } = kf’\left( x \right).\]

    Sum Rule

    Let \(f\left( x \right)\) and \(g\left( x \right)\) be differentiable functions. Then the sum of two functions is also differentiable and

    \[{\left( {f\left( x \right) + g\left( x \right)} \right)^\prime } ={ f’\left( x \right) + g’\left( x \right).}\]

    Let \(n\) functions \({f_1}\left( x \right)\), \({f_2}\left( x \right)\), \(\ldots\), \({f_n}\left( x \right)\) be differentiable. Then their sum is also differentiable and

    \[
    {{\left[ {{f_1}\left( x \right) + {f_2}\left( x \right) + \ldots + {f_n}\left( x \right)} \right]^\prime } }
    = {{f_1}^\prime \left( x \right) + {f_2}^\prime \left( x \right) + \ldots + {f_n}^\prime \left( x \right).}
    \]

    Combining the both rules we see that the derivative of difference of two functions is equal to the difference of the derivatives of these functions assuming both of the functions are differentiable:

    \[{\left( {f\left( x \right) – g\left( x \right)} \right)^\prime } = {f’\left( x \right) – g’\left( x \right).}\]

    We can write the common rule:

    Linear Combination Rule

    Suppose \(f\left( x \right)\) and \(g\left( x \right)\) are differentiable functions and \(a,\) \(b\) are real numbers. Then the function \(h\left( x \right) = af\left( x \right) + bg\left( x \right)\) is also differentiable and

    \[h’\left( x \right) = af’\left( x \right) + bg’\left( x \right).\]

    We add to this list one more simple rule:

    Derivative of the Function \(y = x\)

    If \(f\left( x \right) = x,\) then

    \[f’\left( x \right) = {\left( x \right)^\prime } = 1.\]

    This formula is derived on the Definition of the derivative page.


  • Solved Problems

    Click a problem to see the solution.

    Example 1

    Find the derivative of the function \(y = {x^2} – 5x.\)

    Example 2

    Find the derivative of the function \(y = {\large\frac{{ax + b}}{{a + b}}\normalsize}\), where \(a\) and \(b\) are constants.

    Example 3

    Find the derivative of the function \(y = 2\sqrt x – 3\sin x.\)

    Example 4

    Calculate the derivative of the function \(y = 3\sin x + 2\cos x.\)

    Example 5

    Let \(y = x + \left| {{x^2} – 8} \right|.\) Find the derivative of the function at \(x = 3.\)

    Example 6

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

    Example 7

    Differentiate the function \(y = \large{\frac{{\sin x}}{2}}\normalsize + 3{x^3}.\)

    Example 8

    Find the derivative of the function \(y = \sqrt[3]{x} + 8x.\)

    Example 9

    Calculate the derivative of the function \(y = x\left( {2 + 3x} \right)\) without using the product rule.

    Example 10

    Find the derivative of the function \({y = 3{x^2} – 2\sqrt x + 1.}\)

    Example 11

    Calculate the derivative of the function \(y = \left( {2 – {\large\frac{x}{3}\normalsize}} \right)\left( {{\large\frac{1}{3}\normalsize} + {x^2}} \right).\)

    Example 12

    Calculate the derivative of the function \(y = \left( {x – 1} \right){\left( {x – 2} \right)^2}\) without using the product rule for the derivative.

    Example 13

    Calculate the derivative of the function \(y = \large{\frac{{{x^2} – x – 1}}{x}}\normalsize\) without using the quotient rule.

    Example 14

    Find the derivative of the function \(y = {\large\frac{{{x^2} + 3x + 1}}{x}\normalsize}\) without using the quotient rule for the derivative.

    Example 15

    Find the derivative of the irrational function \(y = 2\sqrt x – 3\sqrt[3]{x}.\)

    Example 16

    Find the derivative of the function \(y = {\left( {x + 4} \right)^3}\) without using the power rule.

    Example 17

    Determine the derivative of \(y = {\left( {x – 1} \right)^4}\) without using the power rule.

    Example 1.

    Find the derivative of the function \(y = {x^2} – 5x.\)

    Solution.

    Using the linear differentiation rules, we have

    \[
    {y’\left( x \right) = {\left( {{x^2} – 5x} \right)^\prime } }
    = {{\left( {{x^2}} \right)^\prime } – {\left( {5x} \right)^\prime } }
    = {{\left( {{x^2}} \right)^\prime } – 5{\left( x \right)^\prime } }
    = {2x – 5 \cdot 1 = 2x – 5.}
    \]

    Example 2.

    Find the derivative of the function \(y = {\large\frac{{ax + b}}{{a + b}}\normalsize}\), where \(a\) and \(b\) are constants.

    Solution.

    \[
    {y’\left( x \right) = {\left( {\frac{{ax + b}}{{a + b}}} \right)^\prime } }
    = {\frac{1}{{a + b}} \cdot {\left( {ax + b} \right)^\prime } = {\frac{a}{{a + b}}.}}
    \]

    Example 3.

    Find the derivative of the function \(y = 2\sqrt x – 3\sin x.\)

    Solution.

    Using the basic differentiation rules, we obtain:

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

    Example 4.

    Calculate the derivative of the function \(y = 3\sin x + 2\cos x.\)

    Solution.

    This expression is a linear combination of two trigonometric functions. The derivative has the following form:

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

    Example 5.

    Let \(y = x + \left| {{x^2} – 8} \right|.\) Find the derivative of the function at \(x = 3.\)

    Solution.

    As \({3^2} – 8 = 1 \gt 0\), the function at the point \(x = 3\) is equivalent to

    \[y\left( x \right) = x + {x^2} – 8.\]

    So, by the sum rule and the power rule (see “Derivatives of Power Functions“),

    \[ {y’\left( x \right) = {\left( {x + {x^2} – 8} \right)^\prime } } = {x’ + {\left( {{x^2}} \right)^\prime } – 8′ } = {1 + 2x + 0 }={ 2x + 1.} \]

    At the point \(x = 3\) the value of the derivative is

    \[y’\left( 3 \right) = 2 \cdot 3 + 1 = 7.\]

    Page 1
    Problems 1-5
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    Problems 6-17