October 22, 2024
The derivative of $\cot x$ is $\boxed{-\csc^2 x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative of […]
October 22, 2024
The derivative of $\csc x$ is $\boxed{-\csc x\cot x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative […]
October 22, 2024
The derivative of $\sec x$ is $\boxed{\sec x\tan x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative […]
October 22, 2024
The derivative of $\tan x$ is $\boxed{\sec^2 x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that \begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*} and \begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*} For a detailed discussion and proof of the derivative of […]
October 22, 2024
The derivative of \( \dfrac{x+1}{x} \) is \( \boxed{-\dfrac{1}{x^2}} \). While we can use the \textbf{quotient rule}, splitting the expression into two terms provides a simpler approach. Method 1: Simplify First (Preferred Method) Rewrite \( \dfrac{x+1}{x} \) as: \[\begin{align*}\frac{x+1}{x} = \frac{x}{x} + \frac{1}{x} = 1 + \frac{1}{x}.\end{align*}\] Now differentiate term-by-term: \[\begin{align*}\frac{d}{dx} \left( 1 + \frac{1}{x} […]
October 11, 2024
The value of \(\cos 180^\circ\) is: $\cos 180^\circ = \boxed{-1}$ Explanation To understand why, let’s look at the unit circle. The cosine of an angle corresponds to the \(x\)-coordinate of the point on the unit circle at that angle. We measure angles by moving counterclockwise from the positive \(x\)-axis. Thus at \(180^\circ\), we’re at the […]
October 8, 2024
Introduction The limit of an absolute value function often involves determining how the function behaves as the input approaches a particular point, especially around points where the expression inside the absolute value changes sign. Key Concept Often limits involving absolute value do not exist. For example: $$\lim_{x\to0}\dfrac{x}{|x|}.$$ Because of the discontinuity on the graph of […]
October 8, 2024
1 Introduction Oftentimes in calculus, we must compute limits that involve trig functions. For example, $$\lim_{x\to{a}}\sin{x}$$ or the limit of any other expression that contains a trig function. This specific limit asks “what value does $\sin{x}$ approach as $x$ approaches $a$?” Keep in mind that some trig limits do not exist. Because of the oscillation […]
October 8, 2024
Polar equations can seem tricky, especially when it comes to taking derivatives. However, the derivative of a polar equation can be taken directly in polar coordinates without needing to convert to rectangular form. This article will guide you through the process with helpful examples. Which Derivative Are You Computing? In polar coordinates, you may want […]
