December 23, 2024
Introduction Exponent rules are the building blocks of simplifying complex calculations. Master them, and math becomes much simpler! These rules help simplify expressions, solve equations, and tackle real-world problems. Let’s break down these laws with clear examples, visuals, and practice problems to ensure they stick. Product Rule Add exponents when multiplying \[\begin{align*} x^m \cdot x^n […]
December 11, 2024
Integration by parts is the inverse of the derivative product rule. It is very useful when $u$-substitution and using standard integration techniques aren’t enough to handle the problem. Integration by parts states that if you have an integral of the form $\displaystyle \int u \: dv$, we may rewrite it as: \[\begin{align*} \int u \: […]
December 11, 2024
There are many different ways to compute integrals. While all of the integration rules are important, they all have their place and can be used in different situations. When it comes to integrating polynomials, the inverse power rule is the most useful technique. The inverse power rule states: \[\begin{align*} \boxed{\int x^n \: dx = \dfrac{x^{n+1}}{n+1} […]
December 6, 2024
Introduction The answer to this question is \[\begin{align*} \boxed{\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}} \end{align*}\] In this article, we’ll learn how to find \(\cos \dfrac{\pi}{4}\), which is the same as \(\cos 45^\circ\). We’ll approach this by using both a right triangle and the unit circle. After that, we’ll go through a practical example where knowing \(\cos \dfrac{\pi}{4}\) […]
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, […]
November 18, 2024
\[\begin{align*} \boxed{\dfrac{d}{dx} \ln\dfrac{1}{x} = -\dfrac1x} \end{align*}\] Solving for the Derivative Notice that $\ln\dfrac1x = -\ln x$. We can see this if we look at $\ln 1 = 0$. We may rewrite the inside: \[\begin{align*} \ln 1 &= \ln\left(\dfrac{x}{x}\right)\\ &= \ln\left(x \cdot \dfrac1x\right)\\ \end{align*}\] We may then apply the product rule for logs to obtain: \[\begin{align*} […]
November 18, 2024
\[\begin{align*} \boxed{\dfrac{d}{dx} \left(\dfrac3x\right) = -\dfrac3{x^2}} \end{align*}\] Solving for the Derivative We may pull the constant, $3$, out of the derivative and focus on $\dfrac{d}{dx} \left(\dfrac1x\right)$. This is the same as $x^{-1}$, so we may apply the power rule: \[\begin{align*} \dfrac{d}{dx} x^n = nx^{n-1} \end{align*}\] Doing this with $x^{-1}$, we get our derivative is equal to […]
November 18, 2024
\[\begin{align*} \boxed{\dfrac{d}{dx} \sec(2x) = 2 \cdot \sec \left(2x\right)\tan \left(2x\right)} \end{align*}\] Solving for the Derivative Applying the chain rule: \[\begin{align*} \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)) \end{align*}\] We may set our $g(x) = 2x$ and our $f(x) = \sec x$. Doing this, we get that \[\begin{align*} \dfrac{d}{dx} \sec(2x) &= \sec \left(2x\right)\tan \left(2x\right)\frac{d}{dx}\left(2x\right)\\ &= \boxed{2 \cdot \sec \left(2x\right)\tan […]
