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
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
Hello everyone! In this article, we will go over how to take the integral of $\dfrac{x-1}{x+1}$. The technique shown in this article can also be used for integrating many fractions of rational equations, so it’s definitely a handy tool to know. Taking the Integral of x-1 over x+1 We will start expressing $\dfrac{x-1}{x+1}$ in a […]
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 […]
October 8, 2024
Introduction Hello everyone! In this article, we will review how to take the derivative of $4^x$. This is a very instructive example for how to take the derivative of an exponential function with a base other than $e$, which is a very important concept to understand. Without further ado, let’s get into it! Taking the […]
October 8, 2024
Introduction Hello everyone! In this blog post, we will discuss how to take the integral of $\sin 2x$. This is important to know, because it touches on two important concepts – integrals of trigonometric functions and $u$-substitution. Integrals of Trig Functions We know that the derivative of $\sin x$ is $\cos x$, and the derivative […]
