December 11, 2024
$$\boxed{\dfrac{d}{dx}\dfrac{x}{2}=\dfrac{1}{2}}$$ To find the derivative of x/2, or $\dfrac{d}{dx}\dfrac{x}{2}$, we’ll use two important derivative rules, the constant multiple rule and the power rule. The first rule we’ll use is the constant multiple rule. It states that $\dfrac{d}{dx}cx=c\cdot\dfrac{d}{dx}$, for any constant $c$. We can, therefore, write $$ \dfrac{d}{dx}\dfrac{x}{2} = \dfrac{d}{dx}(\dfrac{1}{2}\cdot x) = \dfrac{d}{dx}(\dfrac{1}{2}\cdot x). $$ In […]
December 11, 2024
$$\dfrac{d}{dx}\left(\dfrac{2}{x+1}\right)=\boxed{-\frac{2}{(x+1)^2}}$$ In order to compute the derivative of 2/x+1, also written as $\dfrac{d}{dx}\left(\dfrac{2}{x+1}\right)$, we will use the chain rule. First, we can start by rearranging the expression to make the derivative easier to find. Keep in mind, it’s almost always more convenient to factor out a constant and express fractions or square roots as exponents. […]
October 28, 2024
Introduction Before diving into calculus and memorizing formulas, it is important to take a step back and understand what an integral actually means. What is an Integral? An integral, graphically, represents the area under a curve. You might be wondering why it’s useful to know the area under the curve. The reason is that the […]
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
Fundamental Theorem of Calculus: Part one: $$\int_a^bf(x)\:dx=F(b)-F(a)$$ Where $F(x)$ is an antiderivative of $f(x)$ Part two: $$\dfrac{d}{dx}\int^x_af(t)\:dt=f(x)$$ Basic Integration Rules: Constant Rule: $\displaystyle{\int} 0 \:dx =C$ Constant Multiple Rule: $\displaystyle{\int} Cf(x) \:dx = C\displaystyle{\int}f(x) \:dx$ Sum and Difference Rule: $\displaystyle{\int}\left[ f(x)\pm g(x) \right]\:dx = \displaystyle{\int}f(x) \:dx\pm \displaystyle{\int}g(x)\:dx$ Power Rule: $\displaystyle{\int}x^n \:dx= \dfrac{x^{n+1}}{n+1}+C$ Inverse Bounds Rule: […]
