Master Exponent Laws with Ease

ExponentsLeave a comment

Introduction Exponent rules are the building blocks of simplifying complex calculations. Master them, and math becomes much simpler! These rules help simplify expressions, solve equations, and tackle real-world problems. Let’s break down these laws with clear examples, visuals, and practice problems to ensure they stick. Product Rule Add exponents when multiplying \[\begin{align*} x^m \cdot x^n…

Integration by Parts

IntegralsLeave a comment

Integration by parts is the inverse of the derivative product rule. It is very useful when $u$-substitution and using standard integration techniques aren’t enough to handle the problem. Integration by parts states that if you have an integral of the form $\displaystyle \int u \: dv$, we may rewrite it as: \[\begin{align*} \int u \:…

Integral Power Rule

IntegralsLeave a comment

There are many different ways to compute integrals. While all of the integration rules are important, they all have their place and can be used in different situations. When it comes to integrating polynomials, the inverse power rule is the most useful technique. The inverse power rule states: \[\begin{align*} \boxed{\int x^n \: dx = \dfrac{x^{n+1}}{n+1}…

Partial Derivatives

DerivativesLeave a comment

Introduction In calculus, we will often have to work with functions not in one variable, but in two or more. Functions of this type are called $\textbf{multivariable functions}$. Their notation is slightly different. Examples of multivariable functions are $f(x, y) = xy + 1$ and $f(x, y, z) = xy + yz$. The derivative of…

Derivative of x/2

DerivativesLeave a comment

Result: $$\boxed{\dfrac{d}{dx}\dfrac{x}{2}=\dfrac{1}{2}}$$ At first, when you see this problem, you might be tempted to use the quotient rule, since the expression contains a quotient. Indeed, using the quotient rule would work, but here it is not necessary as 2 is a constant. To find the derivative of x/2, or $\dfrac{d}{dx}\dfrac{x}{2}$, we will use the constant…

Derivative of 2/x+1

DerivativesLeave a comment

$$\dfrac{d}{dx}\left(\dfrac{2}{x+1}\right)=\boxed{-\frac{2}{(x+1)^2}}$$ In order to compute the derivative of 2/x+1, also written as $\dfrac{d}{dx}\left(\dfrac{2}{x+1}\right)$, we will use the chain rule. First, we can start by rearranging the expression to make the derivative easier to find. Keep in mind, it’s almost always more convenient to factor out a constant and express fractions or square roots as exponents.…

Understanding cos pi/4 or cos 45 degrees

TrigonometryLeave a comment

Introduction The answer to this question is \[\begin{align*} \boxed{\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}} \end{align*}\] In this article, we’ll learn how to find \(\cos \dfrac{\pi}{4}\), which is the same as \(\cos 45^\circ\). We’ll approach this by using both a right triangle and the unit circle. After that, we’ll go through a practical example where knowing \(\cos \dfrac{\pi}{4}\)…

Integral of $\sec^2x$

IntegralsLeave a comment

\[\begin{align*} \int \sec^2(x) dx = \boxed{\tan(x) + C}. \end{align*}\] To find the indefinite integral of $\sec^2(x)$, let’s first recall that the derivative of $\tan x$ is \[\begin{align*} \dfrac{d}{dx}(\tan(x)) = \sec^2(x) \end{align*}\] The First Fundamental Theorem of Calculus states that \[\begin{align*} f(x) = \dfrac{d}{dx} \displaystyle\int_a^x f(x) dx. \end{align*}\] In other words, derivatives and integrals are inverse…