Chapters

Before we get started, we must identify and define some keywords.

Frac: stands for fractional calculus and it is a branch of mathematical analysis that studies the several different possibilities of defining real number powers.

Derivatives: the derivative of a function* f* is an expression that tells you what the slope of *f* is at any point in the domain of* f*. The derivative of f is a function itself.

Antiderivative: is used to express the inverse derivative, primitive function, primitive integral, or indefinite integral of a function.

Equations are a statement that asserts the equality of two expressions, which are connected by the equals sign "=".

## What Is Differentiation?

According to khana.org Differentiation is the process of finding the derivative, or rate of change, of a function. The objective is to find the rate of change of velocity concerning time (i.e. acceleration) or in other words, the rate of change of ‘x’ concerning ‘y’ on a graph (i.e. the gradient of the curve).

Differentiation, as one of the key branches in calculus, is part of the AP Math curriculum, which means that your teacher is going to dictate a few lessons about its basic rules like Differentiating x to the power of something, along with studying Notation. Know that you can use all of this knowledge to Differentiate the equation of a curve or finding a formula for its gradient.

If you feel calculus is a subject that intimidates you a little, trust that your teacher will be able to explain the process in ways that will be easy for you to understand. But if you want to go the extra mile, a solution might be to check our guide below. We will show you some of the basic Differentiation rules, and some complicated ones if you like the challenge.

## When And Where Might I Need To Use Differentiation?

It is true that differentiation started working to help sailors understand how the planets and stars in our solar system move regarding one another, but with technological advances and navigation systems in place, there is no longer much of a use for Differentiation at sea. So, how is it used in modern society?

Differentiation and Integration are used as tools that help us solve the problems present in the real-world. You will never find a problem that requires you to find something nonexistent or false. For example, derivatives are used by different industries to find the minimum and maximum values of things. These can be things like profit, loss, cost, strength, and material quantity (i.e. for buildings).

*Optimization is the word used for this in the world of industry*

The subject also crops up quite regularly for those working in the field of science and engineering, particularly when looking at the behavior and trends of moving objects.

Nonetheless, even if you’re not working in Physics, Math, Computer Science, or Science, there’s still a chance that you are using Differentiation tools, without even knowing it.

For example, every time someone in business works on how quickly their return can multiply over a specific period of time, or uses a trend to anticipate something happening in the future, they are applying optimization methods, which as we now know, is a form of applying Differentiation and Integration.

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It is true that even though many people use it, it doesn't mean everyone applies it correctly. This is why AP Math is an important part of your education. If you study and learn math at an advanced level, like the AP Math curriculum, you will be able to use the above techniques correctly and use them to your advantage for the future.

## An Introduction To Basic Differentiation Technique

Explicit Differentiation techniques is a term mathematicians use to refer to an equation, in Calculus, for *y* written in terms of *x*. This means it is easy to use basic differentiation techniques to find the derivative and solve the problems.

Nonetheless, some problems present you with other form or type of equation that is complicated to rearrange, for example, when *y* is by itself followed by the *=* sign. In these cases, there is a different approach needed.

The implicit Differentiation method comes in handy when there is a multi-variable equation to work out. But don’t worry, once you learn how to use Explicit Differentiation, this method is going to be easy to learn!

**The Basic Rules Of Simple Differentiation**

#### The Constant Rule

#### The constant rule is one that is referred to as basic Differentiation. The rule works as follows:

\[f (x) = 5\] is a line with a gradient of *0*, therefore it’s derivative is also *0*

As such, if \[f (x) = c (any number)\] then \[f’(x) = 0\]

#### The Power Rule

#### If [latexpage] \[f(x) = x^4\] then to find the derivative you will have to take the power and bring it in front of the *x* before reducing the power by *1*. Therefore, this means that \[f’(x) = 4x^3\] This simple process is to be repeated, so once you comprehend it, it's all you’ll have to do to continue, problem solved! This rule, like the one above, is constant and works for any power, whether it is positive, negative, or a fraction.

Note: \[x to the zero power = 1\]

#### The Constant Multiple Rule

#### Even if the function you are setting out to differentiate begins with a coefficient, this does not affect the process you adopt. Go ahead and differentiate the function using the appropriate rule, with the coefficient staying in place until the final step when you simplify the answer by multiplying by the presented coefficient. So:

If you see \[y = 5x^3\] you can deduce that the derivative of \[x^3\] is \[3x^2\] and therefore that the derivative of \[5(x^3)\] is \[5(3x^2)\] It is only at the end that you simplify in this way: \[5(3x^2)\] equals \[15x^2\] so \[y’ = 15x^2\]

#### The Addition Rule

#### If you come across an addition of terms, you can approach the problem and find the solution by solving the derivative of each term by itself, like this.

Question

What is \[f’(x)\] if \[f(x) = x^6 + x^3 + x^2+ x + 10\]

Answer

\[f’(x) = 6x^5+ 3x^2 + 2x + 1\]

#### The Difference Rule

#### The opposite of an addition, a difference acts just like a subtraction. Therefore, you would solve the equation as follows, (don’t forget you cannot change the subtraction).

Problem

If \[y =3x^5 – x^4 – 2x^3 + 6x^2+ 5x\] then

Answer

\[y’= 15x^4 – 4x^3 – 6x^2 + 12x + 5\]

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## Introduction To Integration Techniques

The term Integration refers to different subtopics and therefore more techniques. In other words, in mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts. Here are the main topics you’ll probably see in this chapter: Integration by Parts, Integrals Involving Trig Functions, Integrals Involving Quadratics, Trig Substitutions, Partial Fractions, Integrals Involving Roots, Integration Strategy, Approximating Definite Integrals, Improper Integrals and, finally, Comparison Tests for Improper Integrals.

**A First Taste Of Integration**

We’ll focus on Integration by Part, given that we can’t explain all of them and that this is a type of problem that most likely many students will come across sometime during their maths studies.

**Integration by Parts**

*Integration by parts or partial integration is a process that finds the integral of a product of two functions*

Remember, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts.

At times, functions appear as a product of another function and need to be integrated. There is a rule that helps in this process, once understood, it can be applied to derive aññ the important Integration by Parts problems.

The technique will allow you to use the formula for Integration by Parts and consequently integrate the products of set functions. Still a little unclear? Check these examples.

The most important thing to remember when faced with an Integration by Parts problem is to rearrange the formula.

For example, if you are present with a problem that looks like this:

\[y = uv\] then \[\frac{\text{d}y}{\text{d}x} = \frac{\text{d}(uv)}{\text{d}x} = u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x}\]

you may rearrange the formula and then integrate in both sides like this:

\[`\int_{}^{} u \frac{\text{d}v}{\text{d}x} dx = \int_{}^{} \frac{\text{d}(uv)}{\text{d}x} dx - \int_{}^{} v \frac{\text{d}u}{\text{d}x}dx`\]

and finally proceed to the final step, which is to simplify the entire formula. Then you would end up with:

\[`\int_{}^{} u \frac{\text{d}v}{\text{d}x} dx = u v - \int_{}^{} v \frac{\text{d}u}{\text{d}x}dx`\]

**Why Do I Need To Learn Integration Techniques?**

Nowadays we have software and computers that can solve these types of problems and use Integration techniques to simplify and solve formulas, so, why should we learn about these methods as students?

Considering that we don’t know what is going to happen in the future, therefore we don’t know if these tools will be used, it seems unnecessary. But, don’t you feel empowered by learning and eventually knowing how the computers you use work and know that you could do it too if one-day technology ceases to exist?

Ok, saying that technology will cease to exist can be a false statement, but we can’t predict the future.

Many people would argue that learning these techniques and methods can be a waste of time since technology does it for us. You need to remember that the people who design the syllabus do it in a way that is going to help you develop your brain and add to your intellect. If there is no practical application of the techniques, take comfort in knowing that at least you are building a capacity to understand complicated mathematical processes as you foster logical intuition.

Whether Calculus is going to be a part of your Bachelor’s or Master’s degree, there is no doubt that you will find everything you learned in AP Math very useful.

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