Derivatives

Application of Derivatives

Derivatives are widely used in mathematics. For example, they can be used to determine the maximum or minimum value of a function, the slope of a curve, or even the point at which a curve inverts. In the following sections, we’ll go into greater detail about each one.

• Calculating a quantity’s change in value
• Calculating an approximation value
• Finding equation of tangent and normal to a curve
• Point of inflection and maxima and minima
• Identifying function’s domain where it increases or decreases

Derivative for Rate of Change of a Quantity

The rate of change formula is one of the most important parts of the application of derivatives. Derivatives can be used to approximate the change in one quantity in relation to the change in another. In this example, let us assume that we’re dealing with a function y = f (x),

which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is

fa + h-fah

With this definition in mind, we can now write the following:

f’a=fa+h-f(a)h

In addition, f(x) at ‘a’ has an instantaneous rate of change of

A very low value of h can now be written as

f’afa+h- fah

or,

f(a+h) ≈ f(a) + f'(a)h

So, in order to find a small change in function, we only need to find the function’s derivative at the given point, and then we can calculate the change. A small change in one variable causes a large change in another, so the derivative estimates the f(x) variable’s change in response to that small change (x).

Approximation value

Using the derivative of a function, you’ll be able to determine the function’s linear approximation at a specific value. After determining the value of a function at a given point, Newton proposed the use of a linear approximation method to find a value that is close to that function. The tangent function has the equation.

 Lx= fa+ f’ax-a

To get the closest value of the function, use the tangent as an approximation to the function’s graph. The linear approximation can be used to estimate the value of √9.1 in this case. Here is the function: fx= y = x.

We’ll calculate √9 and then use linear approximation to calculate √9.1.

We have fx= x, then f’x=12x

Putting a = 9 in Lx= fa+ f’ax-a, we get,

Lx= f9+ f’99.1-9

Lx= 3 + 160.1

Lx≈ 3.0167.

An extremely close approximation to the real-world value of √9.1

Since the function is linearly approximated by using derivatives, we can get a close approximation of it.

Tangent and Normal to a Curve

The derivatives of a function can be used to find the tangent and normal lines of a curve. To determine the slope and equation of the tangent to a curve at a specific point, we can use the derivative of the function’s curve. Lines that only touch the curve once are called tangents. Tangents’ slopes are always equal to their derivative. The derivative of the function (m = f'(x)) yields the slope(m) of the tangent to a curve of the function y = f(x) at the point (x1,y1).

Using the equation to calculate the tangent line’s slope, we can see that m =y-y1x-x1 is the equation of a straight line that passes through this point. Like the equation of the normal line, we can find the curve’s normal equation at a specific point. In other words, the normal line will be perpendicular to the tangent line and vice versa. Hence, we know the slope of the normal line that connects points (x1,y1) on the curve where the function is defined as (y=f(x)).

n = -1/m = – 1/ f'(x)

And by using the equation. 

−1/m=(y−y1)/(x−x1)

We can find the equation of the normal line to the curve.

Maxima, Minima, and Point of Inflection

When attempting to locate a curve’s maximum, minimum, and inflection point, derivatives come in handy. As opposed to the curve’s highs and lows, the point of inflection is the point at which the curve’s character shifts (from convex to concave or vice versa). It is possible to identify the points of maximum and minimum value using the first-order derivative test. As the first step in this test, we find the function’s derivative at the given point and then set it equal to 0 so that the derivative of f'(c) = 0. (here, we have found the slope of the curve is equal to 0, which means it is a line parallel to the x-axis). Assuming the function is defined in the given interval, we can determine whether a given point is a maxima or minima by examining f'(x) values at points to the left and right of the curve and by examining the nature of f'(x).

• When the slope or f'(x) changes its sign from +ve to -ve as we move through point c, we have reached a maximum. And f(c) is the highest value that can be achieved.

• It is called minima when the slope or f'(x) changes its sign from negative to positive as we move through point c. That is why it’s best to use f(c).

• Since f(x) does not change the sign as we move through Point C, this location is referred to as the “Point of Inflection.”

Increasing and Decreasing Functions

We can determine if a function is increasing or decreasing by using derivatives. When it comes to the x-y plane, the increasing function appears to reach the upper right corner, while the decreasing function appears to reach the lower-left corner. Say we have a differentiable function f(x) within limits (a, b). Afterwards, we look at any two points on the function’s curve.

Condition to check increasing, decreasing functions are as follows:

• If at any two points x1 and x2 such that x1 < x2, there exists a relation f(x1) ≤ f(x2), then the given function is increasing function in the given interval, and if f(x1) < f(x2), then the given function is strictly increasing function in the given interval.

• And, if at any two points x1 and x2 such that x1 < x2, there exists a relation f(x1) ≥ f(x2), then the given function is decreasing function in the given interval and if f(x1) > f(x2), then the given function is strictly decreasing function in the given interval.

Conclusion

If we look at mathematics, for example, we can use derivatives to determine the rate at which a quantity changes, as well as the approximation value. Derivatives can also be applied to find the equation for the angle between a line and a curve and the minimum and maximum values of algebraic expression. Following quantities are determined by Application of Derivatives: 

• The rate of change of one quantity with respect to another is determined by using the derivatives method.
• The derivatives of a function can be used to find the tangent and normal lines of a curve.
• The linear approximation of a function at a given value can be found using the derivative of a function.