October 7, 2024
To find the derivative of 1/x, we rewrite it as $x^{-1}$. We may apply the power rule $\dfrac{d}{dx} x^n = nx^{n – 1}$: \[ \begin{align*} \dfrac{d}{dx} x^{-1} &= (-1) \cdot x^{-1-1}\\ \dfrac{d}{dx} x^{-1} &= -x^{-2} \end{align*} \] We get the end result that $\boxed{\dfrac{d}{dx} \dfrac1x = \frac{d}{dx} x^{-1} = -\dfrac1{x^2}}$. In fact, this technique generalizes […]
October 7, 2024
To find the derivative of $\dfrac{1}{1+x}$, we will apply the chain rule $$\dfrac{d}{dx}(f(g(x)) = f'(g(x))g'(x)$$ on the functions $f(x) = \dfrac1x$ and $g(x) = 1+x$. First, we need to find the derivatives of $f(x)$ and $g(x)$: $$f'(x) = \dfrac{d}{dx}(x^{-1}) = -x^{-2} = -\frac{1}{x^2}$$ and $$g'(x) = \dfrac{d}{dx}(x+1) = 1$$ We are now ready to apply […]
October 7, 2024
If you already know that the derivative of $\sin x$ is $\cos x$ and the derivative of $\cos x$ is $-\sin x$, then you can proceed to read this short article. If you wish to learn why $\dfrac d{dx}\sin x = \cos x$ and $\dfrac d{dx}\cos x = -\sin x$, you can refer to the […]
September 4, 2024
The limit of a sequence reveals its long-term behavior. In this article, we will provide a step-by-step guide to determining the limit of a sequence. Step 1: Does the Sequence Converge? The first step to finding the limit of a sequence is to figure out if it converges or not. For example, the sequence \(a_n […]
September 4, 2024
You may remember from your math class that \(\displaystyle\int \cos x \, dx = \sin x + C\), but what happens if it is integral of -cos x? Negative sign is a multiplication by (-1) To get from $\cos x$ to $-\cos x$, we need to multiply it by a negative one: \[ \begin{align*} (-1) […]
September 4, 2024
The Alternating Series Test is a fundamental tool in calculus used to determine whether certain infinite series converge. This test specifically applies to series that alternate in sign, meaning the terms switch between positive and negative. In this article, we’ll discuss the test in detail and walk through three examples of increasing difficulty. The Alternating […]
September 4, 2024
You might have been searching for “Alternate Series Test,” but the proper term is actually “Alternating Series Test.” This test is a useful tool in calculus for determining the convergence of certain types of infinite series, specifically those that alternate in sign. In this article, we’ll explore what the Alternating Series Test is and how […]
September 4, 2024
In this article, we will quickly prove that the harmonic series, \(\sum_{n=1}^{\infty} \frac{1}{n}\), diverges. Does 1/n converge? Proof We will split the terms of the series as follows: \[ 1 = 1 \] \[ \frac{1}{2} = \frac{1}{2} \] \[ \frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \] \[ \frac{1}{5} + \frac{1}{6} + \frac{1}{7} […]
September 4, 2024
Understanding integrals and derivatives of trigonometric functions can be tough. For example, You may have been told that $\dfrac{d}{dx} \sin(x) = \cos(x)$, but not given a good explanation for it. In this article, we will show you with proof how to find the derivatives and integrals of $\cos x$ and $\sin x$ functions. Derivatives of […]
