Derivatives of Simple Functions

Derivatives define the slopes of straight lines or the slopes of tangents drawn to the curves.

Consider a roller coaster: the slope of its path at any given point corresponds to the derivative of the function describing that path.

For a roller coaster's horizontal path, the function can be approximated as a constant. Since the derivative of a constant is always zero, the slope of the path is zero.

If the path is inclined at a 45-degree angle, it can be approximated by a linear function. The derivative of this function is one, showing a constant slope.

But most inclined paths are not at a 45-degree angle. They are scaled differently, where the path can be modeled as a constant coefficient multiplied by x. The derivative of the function is that same constant, showing a uniform slope.

For a curved path that can be approximated by a polynomial function, the slope at any point is given by the derivative of the function. This is found using the power rule: multiply by the exponent and reduce the exponent by one.

If a path can be approximated by an exponential function, the slope of the tangent at any point is given by its derivative at that point, which is proportional to the function itself.