5BSecondDerivatives

Second Derivatives

Second derivatives will be used in curve sketching as a means of understanding the way the line curves and checking the nature of stationary points. They will also be used in solving differential equations.

A second derivative is found by differentiating the derivative of a function. ie Take the derivative twice!

Notation: math \dfrac {d}{dx} \left( \dfrac {dy}{dx} \right) = \dfrac {d^2y}{dx^2} math {pronounced d squared y over dx squared}

Notice that mathematically, you would expect the denominator to be enclosed in brackets to indicate //dx// × //dx//

math f' \left(f'(x) \right) = f''(x) math {pronounced f double-dashed x}

Example 1 Find the second derivative of: math y=3x^4-12x^2+2x math

Differentiate once math \dfrac {dy}{dx} = 12x^3 - 24x + 2 math Differentiate again math \dfrac {d^2y}{dx^2} = 36x^2 - 24 math

Example 2 math \text {Show that } \; y=(2x+3)e^{-3x} math math \text {is a solution to the Differential Equation } \; \dfrac{d^2y}{dx^2}+6\dfrac{dy}{dx} +9y = 0 math

Find the first derivative math \\ \dfrac{dy}{dx} = 2e^{-3x} - 3(2x+3)e^{-3x} \quad \textit{ using product rule} \\ \\ . \quad =(2-6x-9)e^{-3x} \quad \textit { taking out } e^{-3x} \text{ as a common factor} \\ \\ . \quad = (-6x-7)e^{-3x} math

Find the second derivative math \\ \dfrac{d^2y}{dx^2} = -6e^{-3x} - 3(-6x-7)e^{-3x} \quad \textit{ using product rule} \\ \\ . \quad =(-6+18x+21)e^{-3x} \quad \textit { taking out } e^{-3x} \text{ as a common factor} \\ \\ . \quad = (18x+15)e^{-3x} math

Now to prove the equation math \\ LHS =\dfrac{d^2y}{dx^2}+6\dfrac{dy}{dx} +9y \\ \\ . \qquad = (18x+15)e^{-3x} + 6(-6x-7)e^{-3x} + 9(2x+3)e^{-3x} \\ \\ . \qquad = (18x + 15 - 36x - 42 + 18x + 27)e^{-3x} \\ \\ . \qquad = (0)e^{-3x} \\ \\ . \qquad =RHS math

Hence y is a solution to the differential equation, as required. Second Derivatives on the Calculator

Your calculator can do second derivatives. Use the second derivative form, in the **2D** part of the virtual keyboard, select **CALC** on the bottom row.

Example

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