Textbook Question
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = (x2 - 7x - 8) / (x + 1)
3. Techniques of Differentiation
Higher Order Derivatives
11:06 minutesProblem 3.5.79.b
Textbook Question
A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.
Verified step by step guidance1
Start by understanding the differential equation y''(t) + y(t) = 0, where y''(t) is the second derivative of y with respect to t.
Consider the proposed solution y(t) = B cos(t), where B is a constant. We need to verify that this function satisfies the differential equation.
Calculate the first derivative of y(t) = B cos(t) with respect to t. The derivative of cos(t) is -sin(t), so y'(t) = -B sin(t).
Calculate the second derivative of y(t) = B cos(t). The derivative of -B sin(t) is -B cos(t), so y''(t) = -B cos(t).
Substitute y(t) = B cos(t) and y''(t) = -B cos(t) into the differential equation y''(t) + y(t) = 0. This gives -B cos(t) + B cos(t) = 0, which simplifies to 0 = 0, confirming that y = B cos(t) satisfies the equation for any constant B.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Equations
A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a quantity changes over time or space, and can be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs) based on the number of independent variables involved.
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Second-Order Derivatives
In the context of differential equations, a second-order derivative refers to the derivative of a derivative, indicating how the rate of change of a function itself changes. For example, in the equation y''(t), the notation signifies the second derivative of the function y with respect to the variable t, which is crucial for analyzing the behavior of the function.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental periodic functions that arise in various mathematical contexts, including solutions to differential equations. The function y = B cos(t) represents a cosine wave, where B is a constant that affects the amplitude, and it is often used to express solutions to second-order linear differential equations with constant coefficients.
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