October 8, 2024
Introduction One of the most common related rates problems is the ladder problem, which looks at how a ladder slides down a wall, assuming that the ladder always makes a right triangle with the wall. Let’s see how to solve these sorts of problems by working through a simple example. Example Let’s consider a ladder […]
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 […]
October 8, 2024
Basic Derivative Formulas Constant Rule: $\dfrac{d}{dx}C=0$ Constant Multiple Rule: $\dfrac{d}{dx}\Big(Cf(x)\Big)=C\dfrac{d}{dx}f(x)$ Sum and Difference Rule: $\dfrac{d}{dx}\Big(f(x)\pm g(x)\Big)=\dfrac{d}{dx}f(x)\pm\dfrac{d}{dx}g(x)$ Power Rule: $\dfrac{d}{dx}x^n=nx^{n-1}$ Product Rule: $$\dfrac{d}{dx}\Big(f(x)g(x)\Big)=f(x)g'(x)+g(x)f'(x)$$ Quotient Rule: $$\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$$ Chain Rule: $$\dfrac{d}{dx}\Big(f(g(x))\Big)=f'(g(x))g'(x)$$ Where $f(x)$ is the outside function and $g(x)$ is the inside function. Exponential and Logarithmic Derivatives \[ \begin{array}{lll} \dfrac{d}{dx}e^x=e^x && \dfrac{d}{dx}a^x=a^x\ln{a} \\[10pt] \dfrac{d}{dx}\ln{x}=\dfrac{1}{x} && \dfrac{d}{dx}\log_a x=\dfrac{1}{x\ln{a}} \end{array} […]
October 8, 2024
This article will walk you through how to prove the Power Rule and take the derivative of any polynomial. We are going to prove every step of the way. If you don’t want to see the proof and just want a formula, then just know that for any function $f(x) = x^n$, $f'(x) = nx^{n-1}$; […]
October 8, 2024
Hello everyone! In this blog, we will be reviewing everything about derivatives that you need to know for the AP Calculus test, from the definition of a derivative to more advanced topics like implicit differentiation. This blog is for the curious who want to see the proofs of the differentiation rules we all use. Without […]
October 8, 2024
At first, it may appear daunting to calculate the derivative of $\sqrt x$. Indeed, you have not seen anything like this before! You know the power rule, the derivatives of trigonometric functions, like derivative of sine and derivative of cosine. You might even remember the derivatives of tangent, cotangent, secant and cosecant. Unexpectedly, even the […]
October 8, 2024
To find the derivative of $\dfrac{1}{x-1}$, we have to use the power rule and the chain rule. Step 1. Rewrite First, it is easier to rewrite $\dfrac{1}{x-1}$ as $(x-1)^{-1}$. This makes differentiation simpler because we can now apply the power rule directly. Step 2. Power Rule According to the power rule, $\dfrac{d}{dx} x^n = n […]
October 8, 2024
\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (-x) = -1} \end{align*}\] Let’s learn how to find the derivative of the function \( f(x) = -x \). Don’t worry—we’ll keep it simple and easy to understand! What is a Derivative? A derivative tells us how a function changes as the input \( x \) changes. Think of it like measuring how […]
October 8, 2024
\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (x) = 1} \end{align*}\] Let’s learn how to find the derivative of the function $f(x) = x$. It’s simple, we will show every step. What is a Derivative? A derivative tells us how a function changes as the input $x$ changes. Think of it like finding out how fast or slow something is […]
