∫ calculus

1. **Problem statement:** Given functions $$u = \frac{2x - y}{2} x$$

1. Find \(\frac{dy}{dx}\) for the equation \(x^{3} + y^{3} = xy\). First, differentiate both sides with respect to \(x\), remembering that \(y\) is a function of \(x\).

1. **State the problem:** We need to find \(\frac{dy}{dx}\) by differentiating implicitly the equation \(x^3 + y^3 = xy\) with respect to \(x\). 2. **Differentiate each term:**

1. Differentiate implicitly the equation $x^3 + y^3 = xy$ with respect to $x$. 2. Use the product rule on the right-hand side and chain rule on $y^3$. Remember $y$ is a function of

1. **State the problem:** We have the sales function $$S(t) = \frac{125 t^2}{t^2 + 100}$$

1. **State the problem:** Given the population function:

1. We are given the implicit equation $$\frac{x + 3}{y} = 4x + y^{2}$$ and need to find the derivative $$y'(x)$$ implicitly. 2. Rewrite the equation to avoid the fraction:

1. State the problem: Find the derivative $y'(x)$ implicitly defined by the equation $$\sin(xy) = x^2 - 3.$$\n\n2. Differentiate both sides with respect to $x$. Use chain rule on $

1. Find the derivatives of each function using differentiation rules. (i) $f(x) = \sin (2x + 1)$

1. **State the problem:** Find the derivative $y'(x)$ for the implicitly defined function given by $$\sin(xy) = x^2 - 3.$$\n\n2. **Differentiate both sides with respect to $x$: **\

1. The problem asks us to find $$\lim_{x \to \infty} \cos\left(x^{2} + e^{\frac{x!}{2}}\right)$$. 2. Note that $x!$ (factorial of $x$) grows extremely fast as $x$ increases. Theref

1. The problem is to analyze or work with the implicit equation given by $$x e^y - 3 y \sin x = 1$$. 2. This is an implicit relation between $x$ and $y$, which does not easily solv

1. The critical value approach typically involves finding where the derivative of a function equals zero or does not exist, indicating potential maximum, minimum, or inflection poi

1. نُعطى الدالة $$f(x) = x^2 + 5x$$. 2. المطلوب هو إيجاد معادلة المماس للمنحنى حيث يكون المماس عموديًا على مستقيم يميل بزاوية $$\frac{\pi}{4}$$ مع محور $$x$$ السالب.

1. Pernyataan masalah: Cari nilai limit $$\lim_{x\to 2} \frac{x-2}{x^2 + 3 - 10}$$ ketika $$x$$ mendekati 2. 2. Sederhanakan penyebut: $$x^2 + 3 - 10 = x^2 - 7$$.

1. نبدأ بتحديد المعادلة المعطاة للمنحنى: $$f(x) = x^2 + 5x$$. 2. نُريد إيجاد معادلة المماس للمنحنى بحيث يكون المماس عمودياً ويُميل بزاوية $\frac{\pi}{4}$ على محور $x$ السالب.

1. **State the problem:** We want to find the second Taylor polynomial of the function $f(x) = \sqrt[4]{x} = x^{\frac{1}{4}}$ centered at $x = 7$. 2. **Find the derivatives:**

1. The problem is to analyze and understand the function $y = xe^{-2x}$. 2. This function is a product of $x$ and an exponential decay $e^{-2x}$, which means it will grow initially

1. The problem is to understand if I know how to differentiate functions in calculus. 2. Differentiation refers to finding the derivative of a function, which gives the rate at whi

1. **Stating the problem:** Find the derivative of the two given functions using the Chain Rule. ### Part (a)

1. Let's state the problem: We need to show that the function $g(x) = x^3 - 6x^2 + 18x - 2$ is always increasing. 2. To determine whether $g(x)$ is always increasing, we look at it