Step By Step Calculus » 10.7 - Higher Order Derivatives

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Synopsis
Since the derivative of a function is itself a function, one might wonder if there is any special significance attached to taking the derivative of a derivative. The answer is yes. The derivative of a derivative of a function is called the second order derivative of that function, and that nomenclature can be expanded to the nn-th order derivative:
  • f^{(0)}(x)f^{(0)}(x) is the zeroth order derivative of f(x)f(x) (and is, in fact, just f(x)f(x) or yy in Leibniz notation).
  • f^{(1)}(x)f^{(1)}(x) is the first order derivative of f(x)f(x) and is also written f^\prime(x)f^\prime(x) or \dfrac{dy}{dx}\dfrac{dy}{dx}.
  • f^{(2)}(x)f^{(2)}(x) is the second order derivative of f(x)f(x) and is also written f^{\prime\prime}(x)f^{\prime\prime}(x) or \dfrac{d^2y}{dx^2}\dfrac{d^2y}{dx^2}.
  • f^{(3)}(x)f^{(3)}(x) is the third order derivative of f(x)f(x) and is also written f^{\prime\prime\prime}(x)f^{\prime\prime\prime}(x) or \dfrac{d^3y}{dx^3}\dfrac{d^3y}{dx^3}.
  • f^{(n)}(x)f^{(n)}(x) is the n^{\textrm{th}}n^{\textrm{th}} order derivative of f(x)f(x) and is also written \dfrac{d^ny}{dx^n}\dfrac{d^ny}{dx^n}.
Suppose f^{\prime }f^{\prime } is continuous on \left[ a,b\right] \left[ a,b\right] and differentiable on (a,b)(a,b). Then it follows from the Mean Value Theorem that
\[ \begin{array}{ll} \multicolumn{1}{c}{\text{Condition}} & \multicolumn{1}{c}{\text{Result}} \\ f^{(2)}(x)>0\text{ on }(a,b) & f\text{ is \index{concave up}concave up on }\left[ a,b\right] \\ f^{(2)}(x)<0\text{ on }(a,b) & f\text{ is \index{concave down}concave down on }\left[ a,b\right] \end{array} \]