What Is Calculus? A Beginner’s Guide To Limits And Differentiation

Monith Ishanka
25 09 2023
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Calculus is another big branch of mathematics that has many use cases in the real world. In this guide, I will give you a solid understanding of basic calculus concepts that you should know for learning other calculus concepts used in AI and Data Science.

What Is Calculus?

Calculus is a branch of mathematics that deals with the study of change and motion. it provides a framework for understanding how something changes with respect to another thing (how quantities change and interact with each other).

key Concepts in Calculus:

Limits
Derivative
Integral

Limits

limits help us to see what happens to function $f (x)$ when $x$ reaches a certain value. You would think, we can get the answer by solving the equation.

Lets think, $f (x) = \frac{x^{2} - 4}{x - 2}$ , So what happened when $x$ = 2:

$\frac{2^{2} - 4}{2 - 2} = \frac{0}{0}$

We know that $\frac{0}{0}$ can not be defined(Undefined). That’s where limits help us to find what happened to $f (x)$ when $x$ reached 2. Let’s take another approach to understand this,

What happened if x = 2.1:

$f (2.1) = \frac{{2.1}^{2} - 4}{2.1 - 2} = 4.1$

What happened if x = 2.01:

$f (2.01) = \frac{{2.01}^{2} - 4}{2.01 - 2} = 4.01$

You can see when $x$ is close to 2 the output result also gets closer to 4. We can express this by using Limits:

$\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} = \lim_{x \to 2} \frac{(x + 2) (x - 2)}{(x - 2)} = \lim_{x \to 2} (x + 2) = 2 + 2 = 4$

$\lim_{x \to 2} f (x) = 4$

Infinity Limits

What if x reaches to big positive value like $10^{10000000000000}$ or a big negative value like $10^{- 10000000000000}$ Then, the function result will be infinity. Also if the function is divided by a big positive value like $10^{10000000000000}$ then the function result will be 0 or near 0. We can express these two situations using Infinity limits:

If $x$ is close to $\infty$ (big positive value) then the function result also would be a $\infty$ (positive infinity):

$\lim_{x \to \infty} f (x^{2}) = + \infty$

If x is close to $- \infty$ (big negative value) then the function result would be a $\infty$ (big positive infinity) Because ${-∞}^{2} = + \infty$ :

$\lim_{x \to - \infty} f (x^{2}) = + \infty$

Also $\lim_{x \to \infty} f (\frac{1}{x}) = 0$ Because when the divisor is bigger then the dividend result gets closer to 0. like $\frac{1}{10} = 0.1$

Derivatives

The derivative is a function that gives us the slope of the function or the rate at which the function is changing at a particular point. In other words, The derivative of a function represents the rate of change of that function at a particular pointThe process of finding the derivative of some function is called Differentiation. The derivative of a function f(x) is represented as f'(x)

If $y =f(x):$

$f ‘ (x) = y ‘ = \frac{d y}{d x} = \frac{d f}{d x} = \frac{d}{d x} f (x) = D f (x) = D_{x} f (x)$

Where symbols $\frac{d}{d x}$ and $D$ are differentiation operators that indicate the operation of differentiation.

There are several rules and techniques that are commonly used when calculating derivatives of functions in differential calculus. Here are some of the fundamental rules:

Power Rule

$\frac{d}{d x} x^{n} = n \times x^{n - 1}$

For example, the derivative of a function $f (x) = x^{2}$ can be calculated as:

$\frac{d}{d x} x^{2} = 2 \times x^{2 - 1} = 2 x$

Here 2x means that at x, the function f(x) is increasing at a rate of x^2 units per unit change in x. if x = 2 f(x) is increasing at a rate of 4.

We can also use limits to express this:

$f ‘ (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$f (x + h) = (x + h)^{2}$

$f ‘ (x) = \lim_{h \to 0} \frac{(x + h)^{2} - x^{2}}{h} = \lim_{h \to 0} \frac{x^{2} + 2 x h + h^{2} - x^{2}}{h} = \lim_{h \to 0} \frac{2 x h + h^{2}}{h}$

$\lim_{h \to 0} 2 x + h = 2 x$

Constant Multiple Rule

$\frac{d}{d x} C f (x) = C \frac{d}{d x} f (x)$

For example:

$\frac{d}{d x} (9 x^{5}) = 9 [5 \times x^{5 - 1}] = 9 [5 x^{4}] = 45 x^{4}$

Sum Rule

$\frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} f (x) + \frac{d}{d x} g (x)$

Product Rule

$\frac{d}{d x} [f (x) g (x)] = f (x) \frac{d}{d x} g (x) + g (x) \frac{d}{d x} f (x)$

Quotient Rule

$\frac{d}{d x} [\frac{f (x)}{g (x)}] = \frac{g (x) \frac{d}{d x} f (x) - f (x) \frac{d}{d x} g (x)}{{[g (x)]}^{2}}$

These are some of the rules you should apply when calculating the derivative of a function. For a better understanding of derivatives and their outputs. Let’s look at the differentiation with a graph of a function:

Tangent Line – The line that touches the curve at a particular point. (Here C)

Secant Line – The line that touches two points in the curve.

Here, “The slope of the tangent line is equal to the derivative of the function.” So:

$m_{\tan l i n e = f ‘ (x)}$

$m_{\sec a n t l i n e} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

For example, If $f (x) = x^{3}$ and x = 2(C point) then the derivative of the function can be calculated by using the power rule like this:

$f ‘ (x) = 3 x^{2}$

$f ‘ (x) = m_{\tan l i n e} = f ‘ (2) = 3 (2)^{2} = 12$

Here value 12 means that at the point where x = 2, the rate of change of the function is 12, or the slope of the tangent line to the graph at x = 2 would be equal to 12. This indicates that the instantaneous rate of change of the function f(x) at x = 2 is 12 units per unit change in x. In other words, if you were to make an infinitesimally small change in x near x=2, the corresponding change in the function f(x) would be approximately 12 times the change in x

Let’s calculate $m_{\tan l i n e}$ by using the secant line. For that we should have two points related to x which is midpoint, like, $X_{1} = 1 X_{2} = 3$ and $X_{1} = 1.01 X_{2} = 2.99$

$m_{\sec a n t l i n e} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{f (x_{2}) - f (x_{1})}{x_{2} - x_{1}} = \frac{f (3) - f (1)}{3 - 1} = \frac{27 - 1}{2} = 13$

When $x_{1} = 1.90$ and $x_{2} = 2.10$ :

$\frac{f (2.1) - f (1.9)}{2.1 - 1.9} = 12.0$

You can see when two points get closer to 2(C point) the value also gets closer to 12. We can express this using the limits:

$\lim_{x \to 2} \frac{f (x) - f (2)}{x - 2} = \lim_{x \to 2} \frac{x^{3} - 8}{x - 2} = \lim_{x \to 2} \frac{(x - 2) (x^{2} + 2 x + 4)}{x - 2}$

$\lim_{x \to 2} (x^{2} + 2 x + 4) = 2^{2} + 4 + 4 = 12$

Integral

Integral represents the total area under the curve of the function with respect to the independent variable, and the process of finding the Integral of some function is called Integration. Integration is the reverse process of differentiation.

The integral of a function f(x) is denoted as ∫f(x)dx. Where f(x) is the integrand and dx represents the infinitesimal change in the independent variable x.

$\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c$

For example:

$\int 4 x^{3} d x = \frac{4 x^{3 + 1}}{3 + 1} + C = \frac{4 x^{3}}{4} + C = x^{4} + C$

These are the fundamental concepts you must grasp to understand the essential calculus concepts used in the AI field, which I’ll be discussing in Calculus Part-2

Throughout this article, we have learned calculus, and its basic concepts: Limits, Derivatives, Integrals, and use of them. These are the essential knowledge required to learn complex concepts in calculus used in AI and Data Science. And now you can learn those complex concepts confidently.

My Tools

What Is Calculus? A Beginner’s Guide To Limits And Differentiation

What Is Calculus?

Limits

Infinity Limits

Derivatives

Power Rule

Constant Multiple Rule

Sum Rule

Product Rule

Quotient Rule

Integral

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