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## Calculus – Derivatives

The derivative is a central concept of calculus and is known for its many applications in higher mathematics. The derivative of a function at a point can be described in two different ways: geometric and physical. Geometrically, the derivative of a function at a given value of its input variable is the slope of a line tangent to its graph through a given point. This can be found using the slope formula or, given a graph, by drawing horizontal lines toward the input value under investigation. If there is no break or jump at that point in the graph, it is simply the y value corresponding to the given x-value. In physics, a derivative is described as a physical change. It refers to the instantaneous rate of change of an object’s velocity with respect to the short time it takes to travel a certain distance. In relation to this, the derivative of a function at a point in a mathematical representation refers to the rate of change of the value of the output variables as the value of its corresponding input variable approaches zero. In other words, if two carefully chosen values are very close to the given point under question, then the derivative of the function at the point in question is the quotient of the difference between the output values and their corresponding input values, as the divisor approaches. At zero (0).

Precisely, the derivative of a function is a measure of how the function transforms with respect to changes in the values of its input (independent) variables. To find the derivative of a function at a given point, perform the following steps:

1. Select two values, very close to the given point, one to its left and one to its right.

2. Solve for the corresponding output values or y values.

3. Compare the two values.

4. If the two values are equal or nearly equal to the same number, it is the derivative of the function at a certain value of x (the input variable).

5. Using a table of values, if the values of y for those points to the right of the x value in question are approximately equal to the y value approached by the y values corresponding to the chosen input values to the left of x. The value approached is the derivative of the function at x.

6. Algebraically we can find the first derivative function by taking the limit of the differential quotient formula as the denominator approaches zero. Use the derivative function to find the derivative by replacing the input variable with the given value of x.

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