Derivatives

The derivative is a method of measuring how a function gets changed with the changes made in the inputs. Derivative holds an important part in calculus which is a subfield of mathematics. Derivative can be observed on how much a quantity is changed in response to the changes that takes place in another quantity. For instance, if one wants to measure the derivative of the position of an object that is moving with respect to time it would be the object’s instantaneous velocity.

Differentiation:

The best linear approximation of the function is described by the function’s derivative at a particular chosen input value. The method that is followed to compute the rate the change in the independent input ‘x’ which affects the dependent output ‘y’ is referred to as differentiation. The rate at which the change takes place here is referred to as the derivative of y with respect to that of the x. In other words, that the y is a function of x if the dependence of y is upon x. The functional relationship that exist here is denoted in this formula y = f(x) where the function is denoted by the small f. The slope of the graph will be measured by the derivative at each point if the graph of the y will be plotted against the graph of x.

The simplest case can be derived when y become a linear function of x where the graph of y against x will be a straight line. At this point, y= f(x) = mx + b for the real numbers m and b and at this point the slope m is given by the formula

m = (change in y / change in x) = Δy / Δx

Here the symbol Δ is referred to “change in” for which it is used as an abbreviation. This formula is derived as y + Δy = ƒ(x+ Δx) = m (x + Δx) + b = m x + b + m Δx = y + mΔx. Where Δy = m Δx.

Slope of straight line:

In derivative an exact value is given for the slope of a straight line in this way. In other case, if the function f is not linear then this formula is used to find the exact value for the slope of a straight line. Any how the change in y divided by the change in x varies accordingly. In true sense, differentiation is the method which toils to find the exact value for this type of change at a specific given value of x.

Questions:

• What are derivatives?
• What is differentiation?