Calculus

Differentiation of Functions

Differentiation of Functions Logo

Second Derivatives

  • Second Derivative of an Explicit Function

    Let the function \(y = f\left( x \right)\) have a finite derivative \(f’\left( x \right)\) in a certain interval \(\left( {a,b} \right),\) i.e. the derivative \(f’\left( x \right)\) is also a function in this interval. If this function is differentiable, we can find the second derivative of the original function \(f\left( x \right).\)

    The second derivative (or the second order derivative) of the function \(f\left( x \right)\) may be denoted as

    \[{\frac{{{d^2}f}}{{d{x^2}}}\;\text{ or }\;\frac{{{d^2}y}}{{d{x^2}}}\;}\kern0pt{\left( \text{Leibniz’s notation} \right)}\]

    \[{f^{\prime\prime}\left( x \right)\;\text{ or }\;y^{\prime\prime}\left( x \right)\;}\kern0pt{\left( \text{Lagrange’s notation} \right)}\]

    The second derivative has many applications. In particular, it can be used to determine the concavity and inflection points of a function as well as minimum and maximum points.

    In physics, when we have a position function \(\mathbf{r}\left( t \right)\), the first derivative is the velocity \(\mathbf{v}\left( t \right)\) and the second derivative is the acceleration \(\mathbf{a}\left( t \right)\) of the object:

    \[{\mathbf{a}\left( t \right) = \frac{{d\mathbf{v}}}{{dt}} }={ \mathbf{v}^\prime\left( t \right) = \frac{{{d^2}\mathbf{r}}}{{d{t^2}}} }={ \mathbf{r}^{\prime\prime}\left( t \right).}\]

    Other applications of the second derivative are considered in chapter Applications of the Derivative.

    The second derivatives satisfy the following linear relationships:

    \[
    {{\left( {u + v} \right)^{\prime\prime}} = {u^{\prime\prime}} + {v^{\prime\prime}},\;\;\;}\kern-0.3pt
    {{\left( {Cu} \right)^{\prime\prime}} = C{u^{\prime\prime}},\;\;}\kern-0.3pt{C = \text{const}.}
    \]

    Second Derivative of an Implicit Function

    The second derivative of an implicit function can be found using sequential differentiation of the initial equation \(F\left( {x,y} \right) = 0.\) At the first step, we get the first derivative in the form \(y^\prime = {f_1}\left( {x,y} \right).\) On the next step, we find the second derivative, which can be expressed in terms of the variables \(x\) and \(y\) as \(y^{\prime\prime} = {f_2}\left( {x,y} \right).\)

    Second Derivative of a Parametric Function

    Consider a parametric function \(y = f\left( x \right)\) given by the equations

    \[ \left\{ \begin{aligned} x &= x\left( t \right) \\ y &= y\left( t \right) \end{aligned} \right.. \]

    The first derivative of this function is

    \[y’ = {y’_x} = \frac{{{y’_t}}}{{{x’_t}}}.\]

    Differentiating once more with respect to \(x,\) we find the second derivative:

    \[y^{\prime\prime} = {y^{\prime\prime}_{xx}} = \frac{{{\left( {{y’_x}} \right)}’_t}}{{{x’_t}}}.\]


  • Solved Problems

    Click a problem to see the solution.

    Example 1

    Find the second derivative of the function \(y = \large{\frac{x}{{\sqrt {1 – {x^2}} }}}\normalsize.\)

    Example 2

    Find the second derivative of the polynomial function \[y = 3{x^4} – 2{x^3} + 4{x^2} – 5x + 1.\]

    Example 3

    Find the second derivative of the polynomial function \[y = 2{x^5} + 3{x^4} – 4{x^3} + {x^2} – 6.\]

    Example 4

    Find \(y^{\prime\prime},\) if \(y = \cot x.\)

    Example 5

    Find \(y^{\prime\prime}\), if \(y = x\ln x.\)

    Example 6

    Find \(y^{\prime\prime},\) if \(y = {e^{ – {x^2}}}.\)

    Example 7

    Find \(y^{\prime\prime},\) if \(y = x\sqrt {1 + {x^2}}.\)

    Example 8

    Find the second derivative of the function \(y = \sqrt[\large 4\normalsize]{{x + 1}}.\)

    Example 9

    Find \(y^{\prime\prime},\) if \(y = {x^x}.\)

    Example 10

    Calculate \(y^{\prime\prime}\) for the parabola equation \({y^2} = 4x.\)

    Example 11

    Determine the second derivative of the function \(y = \arcsin {\large\frac{{{x^2} – 1}}{{{x^2} + 1}}\normalsize}.\)

    Example 12

    Find the second derivative of the implicitly defined function \({x^2} + {y^2} = {R^2}\) (canonical equation of a circle).

    Example 13

    The function \(y = f\left( x \right)\) is given in parametric form by the equations \[x = {t^3},\;\;y = {t^2} + 1,\] where \(t \gt 0.\) Find \(y_{xx}^{\prime\prime}.\)

    Example 14

    Find the second derivative of the function given by the equation \({x^3} + {y^3} = 1.\)

    Example 15

    The function \(y = f\left( x \right)\) is given in parametric form by the equations\[{x = t + \cos t,\;\;}\kern0pt{y = 1 + \sin t,}\]where \(t \in \left( {0,2\pi } \right).\) Find \(y_{xx}^{\prime\prime}.\)

    Example 16

    Find the second derivative of the function given by the equation \(x + y = {e^{x – y}}.\)

    Example 1.

    Find the second derivative of the function \(y = \large{\frac{x}{{\sqrt {1 – {x^2}} }}}\normalsize.\)

    Solution.

    By the quotient and chain rules, we get

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

    Differentiate again using the power and chain rules:

    \[{y^{\prime\prime} = \left( {\frac{1}{{\sqrt {{{\left( {1 – {x^2}} \right)}^3}} }}} \right)^\prime }={ \left( {{{\left( {1 – {x^2}} \right)}^{ – \frac{3}{2}}}} \right)^\prime }={ – \frac{3}{2}{\left( {1 – {x^2}} \right)^{ – \frac{5}{2}}} \cdot \left( { – 2x} \right) }={ \frac{{3x}}{{{{\left( {1 – {x^2}} \right)}^{\frac{5}{2}}}}} }={ \frac{{3x}}{{\sqrt {{{\left( {1 – {x^2}} \right)}^5}} }}.}\]

    Example 2.

    Find the second derivative of the polynomial function \[y = 3{x^4} – 2{x^3} + 4{x^2} – 5x + 1.\]

    Solution.

    Take the first derivative using the power rule and the basic differentiation rules:

    \[y^\prime = 12{x^3} – 6{x^2} + 8x – 5.\]

    Differentiate once more to find the second derivative:

    \[y^{\prime\prime} = 36{x^2} – 12x + 8.\]

    Example 3.

    Find the second derivative of the polynomial function \[y = 2{x^5} + 3{x^4} – 4{x^3} + {x^2} – 6.\]

    Solution.

    Take the first derivative using the power rule and the basic differentiation rules:

    \[y^\prime = 10{x^4} + 12{x^3} – 12{x^2} + 2x.\]

    The second derivative is expressed in the form

    \[y^{\prime\prime} = 40{x^3} + 36{x^2} – 24x + 2.\]

    Example 4.

    Find \(y^{\prime\prime},\) if \(y = \cot x.\)

    Solution.

    The first derivative of the cotangent function is given by

    \[{y^\prime = \left( {\cot x} \right)^\prime }={ – \frac{1}{{{{\sin }^2}x}}.}\]

    Differentiate it again using the power and chain rules:

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

    Example 5.

    Find \(y^{\prime\prime}\), if \(y = x\ln x.\)

    Solution.

    Calculate the first derivative using the product rule:

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

    Now we can find the second derivative:

    \[
    {y^{\prime\prime} = {\left( {\ln x + 1} \right)^\prime } }
    = {\frac{1}{x} + 0 = \frac{1}{x}.}
    \]

    Page 1
    Problems 1-5
    Page 2
    Problems 6-16