December 11, 2024
Introduction In calculus, we will often have to work with functions not in one variable, but in two or more. Functions of this type are called $\textbf{multivariable functions}$. Their notation is slightly different. Examples of multivariable functions are $f(x, y) = xy + 1$ and $f(x, y, z) = xy + yz$. The derivative of […]
December 11, 2024
Introduction In calculus, derivatives are an extremely useful tool that are used in a variety of problems. In this article, we will specifically learn about taking the derivatives of fractions. Derivatives of Fractions A fraction typically appears as the ratio of two functions in the form $\dfrac{f(x)}{g(x)}$, where \( f(x) \) is the numerator and […]
December 11, 2024
Result: $$\boxed{\dfrac{d}{dx}\dfrac{x}{2}=\dfrac{1}{2}}$$ At first, when you see this problem, you might be tempted to use the quotient rule, since the expression contains a quotient. Indeed, using the quotient rule would work, but here it is not necessary as 2 is a constant. To find the derivative of x/2, or $\dfrac{d}{dx}\dfrac{x}{2}$, we will use the constant […]
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. […]
November 21, 2024
In this article, we will be trying to show that: \[\begin{align*} \dfrac{d}{dx}(\ln(x^3)) = \boxed{\dfrac3x}. \end{align*}\] To start our proof, we will use the Power Rule for logarithms, which tells us $\ln(x^n) = n \ln (x)$. Using this rule, we get that $\ln(x^3) = 3\ln(x)$. Using the fact that $\dfrac{d}{dx} (\ln(x)) = \dfrac{1}{x}$, we have: \[\begin{align*} […]
November 21, 2024
In this article, we will be showing that: \[\begin{align*} \dfrac{d}{dx}(\ln(x^2)) = \boxed{\dfrac2x}. \end{align*}\] To prove this, we will use the Chain Rule. Recalling the formula for the Chain Rule, we find that the Chain Rule states that: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x). \end{align*}\] Where $f(x)$ and $g(x)$ are differentiable functions. Using this formula, we […]
November 21, 2024
We will be finding the derivative of $-\sin x$. We first rewrite this expression as: \[\begin{align*} -\sin x = -1 \cdot \sin x. \end{align*}\] We can remove the $-1$ from the expression because it is a constant. Then, we recall that: \[\begin{align*} \dfrac{d}{dx}(\sin x) = \cos x. \end{align*}\] To learn about how we can prove […]
November 18, 2024
\[\begin{align*} \boxed{\dfrac{d}{dx} \ln x \cdot \ln x = \dfrac{2\ln x}{x}} \end{align*}\] Solving for the Derivative To find the derivative, we will use the Product Rule: \[\begin{align*} \frac{d}{dx} f(x) \cdot g(x) = f(x) \cdot \frac{d}{dx} g(x) + g(x) \cdot \frac{d}{dx} f(x) \end{align*}\] In our case both functions are the same, $f(x) = g(x) = \ln x$, […]
November 18, 2024
\[\begin{align*} \boxed{\dfrac{d}{dx} \dfrac{d}{dx} \ln x = -\dfrac1{x^{-2}}} \end{align*}\] Solving for the Derivative To find the second derivative of $\ln x$, we must first find the first derivative of $\ln x$. The first derivative $\ln x$ is a common derivative, $\dfrac1x$. This is a derivative that you should memorize! Now that we have the first derivative, […]
