December 5, 2024
\[\begin{align*} \int \sec^2(x) dx = \boxed{\tan(x) + C}. \end{align*}\] 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 […]
November 21, 2024
In this article, we will be trying to show that: \[\begin{align*} \dfrac{d}{dx}(\ln(x^3)) = \boxed{\dfrac3x}. \end{align*}\] To start our proof, we will use the Power Rule for logarithms, which tells us $\ln(x^n) = n \ln (x)$. Using this rule, we get that $\ln(x^3) = 3\ln(x)$. Using the fact that $\dfrac{d}{dx} (\ln(x)) = \dfrac{1}{x}$, we have: \[\begin{align*} […]
November 21, 2024
In this article, we will be showing that: \[\begin{align*} \dfrac{d}{dx}(\ln(x^2)) = \boxed{\dfrac2x}. \end{align*}\] To prove this, we will use the Chain Rule. Recalling the formula for the Chain Rule, we find that the Chain Rule states that: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x). \end{align*}\] Where $f(x)$ and $g(x)$ are differentiable functions. Using this formula, we […]
November 21, 2024
We will be finding the derivative of $-\sin x$. We first rewrite this expression as: \[\begin{align*} -\sin x = -1 \cdot \sin x. \end{align*}\] We can remove the $-1$ from the expression because it is a constant. Then, we recall that: \[\begin{align*} \dfrac{d}{dx}(\sin x) = \cos x. \end{align*}\] To learn about how we can prove […]
November 7, 2024
In this article, we are going to prove that the derivative of $e^x$ is equal to $\boxed{e^x}$. To find the derivative of $e^x$, we must first define $e$. The formal definition of $e$ is: \[\begin{align*} e = \lim_{n\to\infty}(1+\dfrac1n)^n \end{align*}\] $e$ is approximately equal to $2.71828$. This value does not need to me memorized, as it […]
November 7, 2024
In this article we will prove the logarithm rule: \[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\] Based on how the logarithms work, $b$ must be positive and not equal to $1$, and $k$ can be any real number. The above rule is saying that if we take the logarithm of a number that is to the $k^{th}$ […]
November 7, 2024
The value of $\log_{10}(1000)$ is $\boxed{\log_{10}(1000) = 3}$. In this article, we will explore why. What does $\log_{10}(1000)$ mean? Let’s start with the expression: \[\begin{align*} \log_{10}(1000) = X \end{align*}\] This equation means that we are looking for a value of \(X\) such that: \[\begin{align*} 10^X = 1000 \end{align*}\] In other words, \(\log_{10}(1000)\) represents the power […]
October 25, 2024
The derivative of $a^x$, where $a$ is a constant is $\dfrac{d}{dx}(a^x) = \boxed{a^x \cdot \ln a}$. Let’s examine why this is true. Finding the Derivative of $a^x$: We will start by rewriting $a^x$ as: \[\begin{align*} a^x = (e^{\ln a})^x = e^{\ln a \cdot x}. \end{align*}\] Now, recall the formula $\dfrac{d}{dx}(e^{nx}) = e^{nx} \cdot n$. In […]
October 7, 2024
The derivative of $e^{nx}$ is $\boxed{n\cdot e^{nx}}$. In this article, we will be exploring why. Recall that the Chain Rule states: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x), \end{align*}\] where $f(x)$ and $g(x)$ are functions. In this case, we let $f(x) = e^x$ and $g(x) = nx$ so that $f(g(x)) = e^{g(x)} = e^{nx}$. The next […]
