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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: 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 the given point. It can be found using the slope formula or, if a graph is given, by drawing horizontal lines to the input value under query. If the graph has no breaks or jumps at this 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 of an object’s velocity with respect to the shortest possible time it takes to travel a certain distance. Relatedly, 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 their 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 point of inquiry is the quotient of the difference between the output values ​​and their input values corresponding, as the denominator approaches. to zero (0).

Precisely, the derivative of a function is a measure of how a function transforms with respect to a change of values ​​in its (independent) input variable. To find the derivative of a function at a given point, follow these 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 equal or will be approximately equal to the same number, then it is the derivative of the function at that given value of x (input variable).

5. Using a table of values, if the y-values ​​of those points to the right of the x-value in question are approximately equal to the y-value to which the y-values ​​corresponding to the input values ​​chosen in the left of x The approximate value is the derivative of the function in 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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