Question

Let \(y=f(x)=x^{3}-4 x\) If \(d x / d t=-2 \mathrm{cm} / \mathrm{sec},\) find \(d y / d t\) at the point where (a) \(x=-3 . \quad\) (b) \(x=1 . \quad\) (c) \(x=4\)

Short answer

The rates of change of the function at the points x=-3, x=1 and x=4 are 2 cm/sec, -2 cm/sec, and -104 cm/sec respectively.

Step-by-step solution

Computing the Derivative of f(x)

The derivative of the function \(f(x) = x^{3} - 4x\) can be calculated using the power rule:\[ f'(x) = 3x^{2} - 4 \]

Apply chain rule for \( dy/dt \) at x = -3

According to the chain rule, \( dy/dt = f'(x) * dx/dt \). At the point x=-3, \[ dy/dt = f'(-3) * -2 \]\[ dy/dt = (3(-3)^{2} - 4)*-2 = 2 \mathrm{cm/sec} \]

Apply chain rule for \( dy/dt \) at x = 1

At the point x = 1,\[ dy/dt = f'(1) * -2 \]\[ dy/dt = (3(1)^{2} - 4)*-2 = -2 \mathrm{cm/sec} \]

Apply chain rule for \( dy/dt \) at x = 4

At the point x = 4,\[ dy/dt = f'(4) * -2 \]\[ dy/dt = (3(4)^{2} - 4)*-2 = -104 \mathrm{cm/sec} \]

Key concepts

Power Rule

The power rule is a fundamental concept in calculus that simplifies the process of differentiating functions that are power functions. It states that if you have a function of the form f(x) = x^n, where n is any real number, the derivative of f with respect to x is f'(x) = nx^{n-1}.

In the given exercise, when computing the derivative of f(x) = x^3 - 4x, we apply the power rule to each term independently. The first term x^3 becomes 3x^2 because according to the power rule, we multiply by the power and then subtract one from the exponent. The second term, being linear, is a special case where n=1, and its derivative is simply the coefficient, which is -4. This simplicity enables us to efficiently find the rate of change of the function at any point.

Derivative of a Function

The derivative of a function represents the rate at which the function's value is changing at a particular point. It is foundational to understanding how quantities vary in relation to one another in calculus. When we find the derivative of a function, we are essentially calculating the slope of the tangent line to the graph of the function at any given point.

For the function f(x) = x^3 - 4x, we computed the derivative f'(x) = 3x^2 - 4 using the power rule. This derivative function now represents the rate of change of f with respect to x. By evaluating this derivative at different values of x, we can determine how fast the value of the function is changing at those particular points.

Rate of Change

The rate of change is an expression of how quickly or slowly a variable quantity changes with respect to another variable. In the context of our problem, the rate of change dy/dt describes how rapidly y, the output of the function f(x), is changing over time, represented by t. The chain rule allows us to link the rate of change of y with respect to x (dy/dx) and the rate of change of x with respect to t (dx/dt).

The chain rule formula is dy/dt = f'(x) * dx/dt, revealing that the overall rate of change of y with respect to time t integrates the rate of change from x to y (f'(x)) and from t to x (dx/dt). This is crucial when variables are interdependent and change with respect to each other, as is often the case in physical phenomena and economics.