Application of Derivatives in Maths

Derivatives play a very important role in the world of Mathematics and also they have various applications beyond mathematics as well i.e., in engineering, architecture, economics, and several other fields. Many proofs of physics and engineering involve applications of derivatives as various physical quantities are the rate change of something over something. For example, velocity is the rate change of displacement, acceleration is the rate change of velocity, etc.

Some of the applications of Derivatives in the field of mathematics are as follows

• Rate Change of Quantities
• Increasing and Decreasing Function
• Approximation
• Monotonicity
• Maxima and Minima
• Tangents and Normal Lines

Rate Change of Quantities

When two quantities are related by some function then a change in one quantity with respect to another quantity is known as a rate change of quantities and it is represented by the derivatives. 

For a function y = f(x), the rate of change of y with respect to x is represented by 

dy/dx = lim _h⇢0[f(x + h) – f(x)]/h 

Where 

• h is the rate change in the value of x,
• f(x + h) – f(x) is the change in the value of function i.e., y, and 
• [f(x + h) – f(x)]/h is the rate of change of y with respect to x. 

Example: For y= 16 – x². Find the rate of change of y at x = 8.

Solution:

The rate of change of y at x = 8 is given by dy/dx at x = 8,

i.e., dy/dx = -2x [Putting  x = 8]

⇒ dy/dx (at x = 8) = -16, 

Hence, -16 is the required answer.

Increasing and Decreasing Function

A function is said to be an increasing function if for function f(x) if we consider two values in its domain x₁ and x₂ such that x₁>x₂, then f(x₁)>f(x₂). In the words of derivatives, we can define the increasing function as the function for which the slope of its graph is positive i.e., for a function f(x), f'(x) > 0, where f'(x) represents the derivative of the given function.

Example: Check whether function f(x) = x² is increasing or not for x > 0.

Solution:

for f(x) =  x²

⇒ f'(x) = 2x 

Now, for x > 0 f'(x) = 2x, is always positive.

Thus f(x) = x², is an increasing function for x > 0.

A function is said to be a decreasing function if for function f(x) if we consider two values in its domain x₁ and x₂ such that x₁>x₂, then f(x₁) < f(x₂). In the words of derivatives, we can define the decreasing function as the function for which the slope of its graph is negative i.e., for a function f(x), f'(x) < 0, where f'(x) represents the derivative of the given function.

Example: Check whether function f(x) = x² is decreasing or not for x< 0.

Solution:

for f(x) =  x²

⇒ f'(x) = 2x 

Now, for x < 0 f'(x) = 2x, is always negative.

Thus f(x) = x², is an decreasing function for x < 0.

Approximation

As derivative is defined as f'(a) = [f(x) – f(a)]/(x-a)

rearranging the above definition, we find the linear approximation formula for any function f(x).

f(x) ≈ f(a) + f'(a)(x – a)

Example: Approximate the value of √0.037 using derivatives.

Solution:

Let’s consider a function f(x) = √x

On differentiating f(x) with respect to x, we get

f'(x) = (1/2) × x^(-1/2) 

As 0.037 can rewritten as 0.04 – 0.003, 

Thus, h = 0.003 

Now, f'(x) = (f(x + h) – f(x)) / h, where h is the change in x.

Thus, -0.003 × (1/2) × 0.04^(-1/2) = f(00.4  -0.003) – f(0.04)

⇒ f(0.037) ≈ 0.1925

⇒ √0.037 ≈ 0.1925

Thus, approximation of √0.037  is 0.1925.

Other than a linear approximation, the formula for the quadratic approximation is given as follows:

f(x) ≈ f(a) + f'(a)(x – a) + (1/2)f”(a)(x – a)²

Where, 

• f(x), f(a), f'(a), and x are defined as in the linear approximation, and 
• f”(a) is the second derivative of the function at a.

Monotonicity

Monotonicity refers to the behavior of a function, specifically how it changes as its input variable changes. A function is said to be monotonically increasing if its output values increase as its input values increase. Similarly, a function is monotonically decreasing if its output values decrease as its input values increase.

More formally, a function f(x) is said to be:

• Monotonically increasing on an interval I if for any x₁, x₂ ∈ I such that x₁ < x₂, we have f(x₁) ≤ f(x₂).
• Monotonically decreasing on an interval I if for any x₁, x₂ ∈ I such that x₁ < x₂, we have f(x₁) ≥ f(x₂).

Maxima and Minima

The tangent to a curve at the point of maxima or minima is a line parallel to the x-axis. The slope of a line parallel to the x-axis is zero. Hence the value of dy/dx at the point of maxima and minima is zero. Now, the steps involved in finding the point of maxima or minima are as follows:

1. Find the derivative of the function. 
2. Equate the derivative with zero to get the critical points.
3. Now find the double derivative of the function.

• If the value of the double derivative at a critical point is less than zero then that point is the point of maxima.
• If the value of the double derivative at a critical point is greater than zero then the point is the point of minima.

Example: Find the local maxima and local minima of the function 2x³ – 21x² + 36x – 20.

Solution:

Let y = 2x³ – 21x² + 36x – 20.

⇒ dy/dx = 6x² – 42x + 36

For Critical point, dy/dx = 0,

⇒ 6x² – 42x + 36 = 0

⇒ x² – 7x + 6 = 0

⇒ x² – (6 + 1)x + 6 = 0

⇒ x² – 6x – x + 6 = 0

⇒ x = 6, 1.

Thus, the critical points are 6 and 1.

Now, d²y/dx² = 12x – 42

Putting x = 6.

d²y/dx² = 12 × 6 – 42 = 30 > 0 

Hence, 6 is a point of minima.

Minimum value is 2 × 216 – 21 × 36 + 36 × 6 – 20 = -128

Putting x=1.

d²y/dx² = 12-42 = -30 < 0 

Hence, 1 is apoint of maxima.

Maximum value is 2 – 21 + 36 – 20 = -3. 

Tangent and Normal

A line that touches a curve at a point but does not pass through it, is called the tangent to the curve at that point. A normal is a line that is perpendicular to a tangent. The equation of a tangent to a curve is shown in the graph below,

Let y = f(x) be a single-valued function and QRTP be the curve of the function. RT is a chord or a straight line. Coordinates of R = (x, y) and coordinates of T = (x + ∆x, y + ∆y). Slope of a line = m = (y₂ – y₁) / (x₂ – x₁), 

Slope of RT = (y+∆y – y) / (x+∆x – x)

⇒ Slope of RT = ∆y / ∆x ⇢ (1)

Now, the equation of the chord RT, Y – y = (Slope of RT) × (X – x), [x and y are coordinates of R.]

Y – y = (∆y / ∆x) × (X – x) ⇢ (2), 

The slope of RT = ∆y / ∆x.

Now, if the point T gradually moves towards R and in time coincides with R then the chord RT transforms itself to tangent MRLN. This happens when ∆x tends to zero. Therefore equation 2 changes to:

Equation of tangent MN = lim ∆x ⇢ 0   (Y – y) = (∆y / ∆x) × (X – x), This can be written as,

(Y – y) = lim_{∆x ⇢ 0} (∆y / ∆x) ×  (X – x).

According to the definition of derivatives,

dy/dx = lim_{∆x ⇢ 0} (∆y / ∆x).

Therefore the equation of tangent MN: (Y – y) = dy/dx × (X – x). This is how we can use the concept of differentiation to find the equation of a tangent to a curve. The normal to a curve is perpendicular to the tangent to the curve.

Note: If two lines are parallel to each other, they both have the same slope. If two lines are perpendicular to each other, the multiplication of their slopes is equal to -1.

As we know that a normal curve is perpendicular to the tangent, therefore,

The slope of normal × Slope of tangent = -1.

Let the slope of normal be m. We know that the slope of tangent = dy/dx. Therefore,

m × dy/dx = -1

⇒ m = – dx/dy

Therefore the equation of normal to the curve at R is given by,

(Y – y₁) = (-dx/dy) × (X – x₁)

Where, -dx/dy is the slope of normal at (x₁, y₁) 

Hence, the concept of the derivatives is used in finding the equations of both the tangent and the normal to a curve at a given point.

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. This makes Derivatives very useful in various fields. For example, derivatives help in understanding motion, growth, and change in physical, economic, and engineering systems. They are used to find rates of change, slopes of curves, and to solve optimization problems. By understanding derivatives, we can predict how things will change and make better decisions based on this information.