## Instantaneous Rate of Change at x=a

### The instantaneous rate of change at x=a refers to the rate at which a function’s value is changing with respect to the input variable, at a specific point on the function

The instantaneous rate of change at x=a refers to the rate at which a function’s value is changing with respect to the input variable, at a specific point on the function. Mathematically, it can be represented as the derivative of the function evaluated at x=a.

The derivative of a function measures the steepness of the function’s graph at different points. At any given point on the graph, the derivative represents the slope of the tangent line to the graph at that point. This slope describes how fast the function is changing at that specific point.

To find the instantaneous rate of change at x=a, you can start by differentiating the function to obtain its derivative. Then, substitute x=a into the derivative expression and evaluate the resulting expression. The value you obtain will be the instantaneous rate of change of the function at x=a.

Here is an example to illustrate the process:

Let’s say we have a function f(x) = 3x^2 + 2x – 1, and we want to find the instantaneous rate of change at x=2.

Step 1: Differentiate the function f(x)

f'(x) = 6x + 2

Step 2: Substitute x=2 into the derivative expression

f'(2) = 6(2) + 2 = 14

Therefore, the instantaneous rate of change of f(x) at x=2 is equal to 14. This means that at x=2, the function is changing at a rate of 14 units per unit change in the input variable.

It’s important to note that the instantaneous rate of change can vary at different points on a function. By evaluating the derivative at different values of x, you can determine how the rate of change varies throughout the function.

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