Discovering the Instantaneous Rate of Change: Calculating Derivatives and Tangent Line Slopes

Instantenous Rate of Change

The instantaneous rate of change, also known as the derivative, measures how a function is changing at a specific point

The instantaneous rate of change, also known as the derivative, measures how a function is changing at a specific point. It allows us to determine the slope of the tangent line to the curve at that particular point.

To calculate the instantaneous rate of change of a function, we need to find its derivative. The derivative of a function f(x) is denoted as f'(x) or dy/dx and is defined as the limit of the average rate of change as the interval approaches zero.

If we have a function f(x), the instantaneous rate of change at a particular point x = a can be calculated using the following formula:

f'(a) = lim (h→0) [f(a + h) – f(a)] / h

Here, h represents the small change in x as it approaches zero. By taking the limit of this expression as h approaches zero, we can find the slope of the tangent line at the point (a, f(a)).

To illustrate this concept, let’s consider an example:

Given the function f(x) = 2x^2, we want to find the instantaneous rate of change at x = 3.

Step 1: Find the derivative of the function.
To find the derivative of f(x) = 2x^2, we can use the power rule of differentiation. The power rule states that if we have a function of the form f(x) = ax^n, then its derivative is given by f'(x) = nax^(n-1).

Applying the power rule to our function, we have:
f'(x) = 2 * 2x^(2-1)
= 4x

So, the derivative of f(x) = 2x^2 is f'(x) = 4x.

Step 2: Evaluate the derivative at x = 3.
Substituting x = 3 into the derivative, we get:
f'(3) = 4(3)
= 12

Therefore, the instantaneous rate of change of f(x) = 2x^2 at x = 3 is 12. This means that at x = 3, the function is changing at a rate of 12 units per unit change in x.

Remember that the instantaneous rate of change represents the slope of the tangent line to the curve at a specific point. It provides valuable information about how the function behaves locally around that point.

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