## Derivative of a horizontal tangent line?

### When a tangent line is horizontal, it means that its slope is equal to zero

When a tangent line is horizontal, it means that its slope is equal to zero. In calculus, the derivative represents the rate of change of a function at a specific point.

To find the derivative of a function at a point where the tangent line is horizontal, we need to determine the slope of the tangent line. However, since the slope is zero, it implies that the function does not change at that specific point.

Mathematically, if the tangent line to a function is horizontal, then the derivative of the function at that point is equal to zero.

For example, let’s take the function f(x) = x^2. To find the derivative of this function, we can use the power rule, which states that if f(x) = x^n, then the derivative of f(x) is n*x^(n-1).

In this case, the derivative of f(x) = x^2 is f'(x) = 2x. If we want to find the derivative at a point where the tangent line is horizontal, we set the derivative equal to zero:

2x = 0

Dividing both sides by 2, we get:

x = 0

So, at x = 0, the tangent line to the function f(x) = x^2 is horizontal, and the derivative at that point is zero.

In summary, if the tangent line to a function is horizontal, the derivative of the function at that point is equal to zero.

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