Calculus question

Question

Why is the derivative of x^3 equal to 3x^2?

Shouldn't the derivative be a linear function or a constant since we're dealing with the slope of a tangent line at a point and why is the derivative a quadratic like 3x^2.

unlike x^2 .the derivative is 2x and this is a linear function. The derivative,2x, is a linear function of the tangent line at a point.

Isn't the derivative equal to the slope of a tangent line at a point . and the tangent line has a linear slope or a constant.

Answer 1

You are confusing the derivative at a point with the derivative function.

The derivative at a point is the slope of the tangent line at a specific x value, which of course is a constant value.

The derivative function is the slope of ALL tangent lines at ALL values of x. Clearly the slopes of all of the tangent lines for the function f(x) = x^3 are not constant. Those slopes vary. Hence the derivative function is not a constant and will vary with x.