The slopes of the tangent lines to the graph of the function f(x) increase as x increases.

At what rate do the slopes of the tangent lines increase?
f(x) = x2 – 1

Thanks for the help.. really need it.

To find the rate at which the slopes of the tangent lines increase for the given function, we need to find the derivative of the function f(x) = x^2 - 1.

The derivative of a function measures the rate of change of the function at any given point. In this case, the derivative will give us the slope of the tangent line to the graph of f(x) at each point.

To find the derivative of the given function, 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, where a is a constant and n is any real number, then the derivative of f(x) is given by:

f'(x) = n * ax^(n-1)

Applying this rule to our function f(x) = x^2 - 1, we find:

f'(x) = 2 * x^(2-1) = 2x

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

Now, since we are interested in the rate at which the slopes of the tangent lines increase, we need to find the derivative of the derivative, which is also known as the second derivative. In other words, we need to find the derivative of f'(x) = 2x.

Using the power rule again, we find:

f''(x) = 2 * 1 * x^(1-1) = 2

Therefore, the second derivative of f(x) is f''(x) = 2.

Since the second derivative is a constant value of 2, it tells us that the rate at which the slopes of the tangent lines increase is a constant rate of 2 units per increase in x.

So, the slopes of the tangent lines to the graph of the function f(x) = x^2 - 1 increase at a rate of 2.