December 11, 2024

Meaning of the Logarithm: A Quick Guide

The logarithm function is an important function that arises from finding the inverse function of an exponential. More specifically, if \( a^y = x \), then \( y = \log_a(x) \). Definition The logarithm of \( x \) to base \( a \) is: \[\begin{align*} \log_a(x) = y \quad \text{if and only if} \quad a^y…

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Integration by Parts

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 \:…

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Integral Power Rule

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}…

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Partial Derivatives

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…

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Quotient Rule: Taking the Derivatives of Fractions

Introduction In calculus, derivatives are an extremely useful tool that are used in a variety of problems. In this article, we will specifically learn about taking the derivatives of fractions. Derivatives of Fractions A fraction typically appears as the ratio of two functions in the form $\dfrac{f(x)}{g(x)}$, where \( f(x) \) is the numerator and…

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Derivative of x/2

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…

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Derivative of 2/x+1

$$\dfrac{d}{dx}\left(\dfrac{2}{x+1}\right)=\boxed{-\frac{2}{(x+1)^2}}$$ 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.…

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