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Calculus – Derivatives
The derivative is the 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 some value of its input variable is the slope of the tangent line to its graph passing through the given point. It can be found by using the slope formula or, if given a graph, by drawing horizontal lines to the input value in question. If the graph has no break or jump at that point, it is just 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 of an object’s speed relative to the shortest possible time it takes to travel a certain distance. Related to this, the derivative of a function at a point in a mathematical view refers to the rate of change in the value of the output variables as the values of its corresponding input variables approach zero. In other words, if two carefully chosen values are very close to the given point in question, then the derivative of the function at the survey point is the quotient of the difference between the output values and their corresponding input values, as the denominator approaches zero (0).
Specifically, the derivative of a function is a measure of how a function transforms with respect to a change in values in its (independent) input variable. To find the derivative of a function at a certain point, follow these steps:
1. Choose two values, very close to the given point, one to its left and the other to its right.
2. Find the corresponding output values or y-values.
3. Compare the two values.
4. If the two values are the same or will be approximately equal to the same number, then this is the derivative of the function at that certain value of x (input variable).
5. Using a table of values, if the y-values for the points to the right of the x-value in question are approximately equal to the y-value approximated by the y-values corresponding to the chosen input values to the left of x. The approximate value is the derivative of the function at x.
6. Algebraically, we can first find the derivative function by taking the limit of the difference 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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