Higher-Order Derivatives

Learning Outcomes

  • Explain the meaning of a higher-order derivative

The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of [latex]y=f(x)[/latex] can be expressed in any of the following forms:

[latex]f''(x), \, f'''(x), \, f^{(4)}(x), \cdots ,f^{(n)}(x)[/latex]

 

[latex]y'', \, y''', \, y^{(4)}, \cdots ,y^{(n)}[/latex]

 

[latex]\dfrac{d^2y}{dx^2}, \, \dfrac{d^3y}{dx^3}, \, \dfrac{d^4y}{dx^4}, \cdots,\dfrac{d^ny}{dx^n}[/latex]

 

It is interesting to note that the notation for [latex]\frac{d^2y}{dx^2}[/latex] may be viewed as an attempt to express [latex]\frac{d}{dx}\left(\frac{dy}{dx}\right)[/latex] more compactly. Analogously,

[latex]\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right)=\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)=\frac{d^3y}{dx^3}[/latex]

Example: Finding a Second Derivative

For [latex]f(x)=2x^2-3x+1[/latex], find [latex]f''(x)[/latex].

Try It

Find [latex]f''(x)[/latex] for [latex]f(x)=x^2[/latex].

Watch the following video to see the worked solution to the above Try It.

Example: Finding Acceleration

The position of a particle along a coordinate axis at time [latex]t[/latex] (in seconds) is given by [latex]s(t)=3t^2-4t+1[/latex] (in meters). Find the function that describes its acceleration at time [latex]t[/latex].

Watch the following video to see the worked solution to Example: Finding Acceleration.

Try It

For [latex]s(t)=t^3[/latex], find [latex]a(t)[/latex].