Applications of Geometrical and Physical Interpretation of a Derivative

Introduction

The algebraic process of calculating derivatives is known as differentiation. The slope or gradient of a particular graph at any given position is the derivative of a function. The tangent’s value drawn to that curve at any given location is the gradient of that curve. The curve’s gradient varies at different positions along the axis for non-linear curves. As a result, calculating the gradient in such situations is challenging.

It’s also known as a property’s change about another property’s unit change.

Consider the function f(x) as a function of the independent variable x. The independent variable x is caused by a tiny change in the independent variable Δx. The function f(x) undergoes a similar modification Δf(x). Ratio  Δf(x)Δx is a measure of f(x) of change in relation to x.

As Δx approaches zero, the ratio’s limit value is

limΔx0f(x+Δx)-f(x)Δx  is known as the first derivative of the function f(x).

Geometrical Interpretation of a Derivative

Let us understand how to interpret a derivative geometrically.

Tangent and Normal to a Curve

We can use the derivatives of a function to find the tangent and normal lines of a curve. The derivative of the function’s curve helps determine the slope and equation of the tangent to a curve at a specific point. 

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−y1)/(x−x1) is the equation of a straight line that passes through this point. Hence, we know the slope of the normal line that connects point (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)

Thus, we can find the equation of the normal line to the curve.

Physical Interpretation of a Derivative

Derivative for Rate of Change of a Quantity

The rate of change formula is one of the most important applications of derivatives. We can use derivatives to approximate the change in one quantity in relation to the change in another. In this example, let us assume that we are dealing with a function y = f (x).

If the interval [a, a+h] defines this function, its average rate of change in the given interval is

(f(a + h)-f(a))/h

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

f'(a)=h0 f(a+h)−f(a)/h

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

A very low value of h can now be written as

f'(a) ≈ (f(a+h) − f(a))/h

or,

f(a+h) ≈ f(a) + f'(a)h

So, to calculate a small change in function, we only need to find the function’s derivative at the given point. A small change in one variable causes a significant 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 will be able to determine the function’s linear approximation at a specific value. After assessing the value of a function at a given point, Newton proposed using a linear approximation method to find a value that is close to that function. The tangent function has the equation.

 L(x) = f(a) + f'(a)(x−a)

To get the closest value of the function, use the tangent as an approximation to the function’s graph. 

For example, we can use the linear approximation to estimate the value of √9.1 in this case. Here is the function: 

f(x) = y = √x.

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

We have f(x) = √x, then f'(x) = 1/(2√x)

Putting a = 9 in L(x) = f(a) + f'(a)(x−a), we get,

L(x) = f(9) + f'(9)(9.1−9)

L(x) = 3 + (1/6)0.1

L(x) ≈ 3.0167

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

Thus, we can get a close approximation of a function by its linear approximation using derivatives.

Conclusion

We can use derivatives to determine the rate at which a quantity changes and the approximation value from a mathematical perspective. We can also apply them to find the equation for the angle between a line and a curve and the minimum and maximum values of algebraic expression. The applications of the geometrical and physical interpretation of derivatives include: 

• Determining the rate of change of one quantity with respect to another using the derivatives method.
• Using the derivative of a function to find the tangent and normal lines of a curve.
• Utilize the derivative method to find a function’s linear approximation at a given value.