Calculus – Derivatives

Lisa R. Parker

Derivative is the central concept of Calculus and is known for its numerous applications to higher Mathematics. The derivative of a function at a point can be described in two different ways: geometrical and physical. Geometrically, the derivative of a function at a certain value of its input variable is the slope of the line tangent to its graph through the given point. It can be found by using the slope formula or if given a graph, by drawing horizontal lines toward the input value under inquiry. If the graph has no break or jump at that point, then it is simply the y value corresponding to the given x-value. In Physics, the derivative is described as a physical change. It refers to the instantaneous rate of change in the velocity of an object with respect to the shortest possible time it takes to travel a certain distance. In relation thereof, the derivative of a function at a point in a Mathematical view refers to the rate of change of the value of the output variables as the values of its corresponding input variables get close to 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 of inquiry is the quotient of the difference between the output values and their corresponding input values, as denominator gets close to zero (0).

Precisely, the derivative of a function is a measurement of how a function transforms with respect to a change of values in its input (independent) variable. To find the derivative of a function at a certain point, do the following steps:

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

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

3. Compare the two values.

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

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

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

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