Differential Calculus

Differential calculus is a fundamental branch of mathematics that focuses with the study of rates of change and the slopes of curves. It's a powerful tool used to analyze the behavior of functions and understand how things change over time or space. This area of calculus is essential not only in mathematics but also in various fields like physics, engineering, economics, and computer science.

1.0What is Differential Calculus?

Differential calculus centered around the concept of derivatives, which measure the rate at which a quantity changes. The derivative of a function provides us with valuable information about the function’s slope or its instantaneous rate of change at any given point. This is particularly useful in many real-world problems where we need to understand how variables are related and how one variable changes in response to another.

In simpler terms, differential calculus meaning revolves around the idea of finding how much a quantity changes as another quantity changes infinitesimally. For example, in physics, differential calculus helps in understanding velocity, acceleration, and other concepts that describe motion.

2.0The Basics of Differential Calculus

At the heart of basic differential calculus is the idea of the derivative. The derivative helps us understand how a function changes when its input (usually represented as xx) changes by a very small amount.

Imagine you have a curve, and you want to know how steep it is at a specific point. The derivative tells you the slope of the curve at that point—how quickly the function's value changes as the input x changes.

To find the derivative of a function f(x) at a point x = a, we use a special formula. The formula looks like this:

$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$

Let’s break it down:

• f(a + h) - f(a): This measures how much the function’s output changes when the input changes by a small amount, h.
• h: This is the small change in the input xx.
• The limit: The "lim" part means that we want to look at how this rate of change behaves as h gets closer and closer to zero, meaning we’re looking at a very tiny change in the input.

In simpler terms, what this formula is saying is: The derivative tells us how much the function changes for an extremely small change in its input. And when we calculate it, we get the slope of the function at a specific point.

3.0Differential Calculus and Integral Calculus: A Relationship

One of the biggest breakthroughs in mathematics was realizing that differential calculus and integral calculus are closely connected. These two areas were developed separately by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 1600s. Together, they form the foundation of calculus, which has had a huge impact on science and engineering.

Differential calculus is all about rates of change (which we measure with derivatives), while integral calculus focuses on the accumulation of quantities (which we calculate using integrals). The fundamental theorem of calculus links these two ideas: it says that the integral of a function over an interval represents the total area under its curve, and the derivative of that integral will give back the original function.

In simple terms, differential calculus looks at how things change, and integral calculus looks at how things add up.

4.0Differential Calculus Example

Example 1: Find the derivative of the function $f(x)=x^2$. Also find the value for x = 3. 

Solution: 

Using basic rules of differentiation, we can compute:

f'(x) = 2x  

This means that the slope of the function _f(x)=x^2. at any point x is twice the value of x. For instance, at x = 3, the slope of the curve is:

f'(3) = 2(3) = 6 

This indicates that at x = 3, the function is increasing at a rate of 6 units per unit change in x.

Example 1: Find the derivative of the following function:

$f(x)=3x^3-5x^2+2x-7.$

Solution: 

We are asked to find f'(2), so we will apply the limit definition of the derivative:

$f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{f(2+h)-f(2)}{h}$

First, we need to calculate f(2 + h), which means plugging 2 + h into the function .

$f(x)=3x^3-5x^2+2x-7.$

$f(2+h)=3(2+h{)}^3-5(2+h{)}^2+2(2+h)-7.$

Now, let's expand each term.

$(2+h{)}^3=8+12h+6h^2+h^3$

$(2+h{)}^2=4+4h+h^2$

2(2 + h) = 4 + 2h

Substitute these into the equation for f(2 + h):

$f(2+h)=3(8+12h+6h^2+h^3)-5(4+4h+h^2)+4+2h-7$

Now simplify:

$f(2+h)=24+36h+18h^2+3h^3-20-20h-5h^2+4+2h-7$

$f(2+h)=1+18h+13h^2+3h^3$

Now, substitute x = 2 into the original function:

$f(2)=3(2{)}^3-5(2{)}^2+2(2)-7$

Now, we can calculate the difference quotient $\frac{f(2+h)-f(2)}{h}$:

$\frac{f(2+h)-f(2)}{h}$

$\frac{(1+18h+13h^2+3h^3)-1}{h}$

$\frac{18h+13h^2+3h^3}{h}$

Now, simplify the expression by dividing each term by h:

$18+13h+3h^2$

Finally, take the limit as :

$f’(2)=\underset{h\to 0}{\lim }(18+13h+3h^2)$

As h approaches 0, the terms involving h vanish:

f'(2) = 18

This means that the slope of the curve at x = 2is 18.

5.0Applications of Differential Calculus

Differential calculus has vast applications in many fields:

Physics: It helps in understanding motion, acceleration, and forces.

Economics: It is used to model cost functions, supply-demand curves, and maximize profits.

Engineering: Differential equations model the behavior of electrical circuits, fluid flow, and structural mechanics.

Biology: Differential calculus is used in modeling population growth, spread of diseases, and enzyme kinetics.

6.0Solved Examples 

Q1. What is the power rule in differentiation?

Answer:

The power rule is one of the basic rules of differentiation. It states that if a function is of the form $f(x)=x^n$ , where n is a constant, then its derivative is given by:

$f’(x)=n.x^{n-1}$

For example, the derivative of $f(x)=x^3$ is $f’(x)=3x^2$