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Calculus – Derivatives
The derivative is a central concept in calculus and is known for its numerous applications in higher mathematics. The derivative of a function at a point can be described in two different ways: geometrically and physically. Geometrically, the derivative of a function at a given value of its input variable is the slope of the tangent line to its graph through a given point. This can be found using the slope formula or, in the case of a graph, by drawing horizontal lines in the direction of the input value under query. 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, a derivative is described as a physical change. It refers to the instantaneous rate of change of an object’s velocity relative to the shortest possible time required to travel a given distance. In this regard, the derivative of a function, in a mathematical view, refers to the rate at which the value of the output variables at a point changes as the values of its corresponding input variables approach zero. In other words, when two carefully chosen values are very close to a given point, the derivative of the function is the quotient of the difference between the output values at the query point and their corresponding input values as the denominator approaches zero (0).
Specifically, the derivative of a function is a measure of how the function transforms with respect to changing values of its (independent) input variable. To find the derivative of a function at a given point:
1. Pick two values that are very close to the given point, one to the left and one to the right.
2. Solve for the corresponding output values or y values.
3. Compare the two values.
4. If two values are the same or approximately equal to the same number, then this is the derivative of the function at some value of x (the input variable).
5. Using a table of values, if the y values 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 selected input values to the left of x. The value approached is the derivative of the function x.
6. Algebraically, we can first find the derivative function by taking the limit of the intermediate 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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