∫ calculus

1. **Problem Statement:** We are given the graph of the first derivative of a function $f$, denoted $f'(x)$. We need to determine which of the given graphs may represent the origin

1. The problem states that $f'(3)$ is undefined and $f''(x) > 0$ for all $x \neq 3$. This means the function has a critical point (likely a cusp or sharp corner) at $x=3$, and is c

1. Problem statement: Given a function $f(x)$ and its inverse function $g(x) = f^{-1}(x)$, determine the shape (increasing/decreasing and convexity) of the function $g$ based on th

1. Given the equation $$\sqrt{x} + y = 9 + x^{2}y^{2}$$, we need to find $$\frac{dy}{dx}$$. 2. Differentiate both sides with respect to $$x$$.

1. **State the problem:** Differentiate the equation $$2 + 6x = \sin(xy^{2})$$ implicitly to find $$\frac{dy}{dx}$$. 2. **Differentiate both sides:**

1. We are given the implicit equation $$e^{x^2 y} = x + y$$ and need to find $\frac{dy}{dx}$ using implicit differentiation. 2. Differentiate both sides with respect to $x$.

1. **Stating the problem:** Given the implicit equation $$xy + 2x + 3x^2 = -8$$ we need to: (a) Find $y'$ (the derivative of $y$ with respect to $x$) implicitly.

1. We are asked to find \( \frac{d y}{d x} \) for \( y = \cos \left( \frac{1 + x^{2}}{1 - x^{2}} \right) \). Step 1: Let \( u = \frac{1 + x^{2}}{1 - x^{2}} \).

1. **Problem Statement:** Find the coordinates of the stationary points for the curve y = \frac{9}{2x-5} + 2x - 5.

1. Let's interpret the request as solving a problem involving "step 20" of a derivation process. 2. Since no specific problem was provided, I'll demonstrate a detailed derivation p

1. **Problem (a):** Given the curve $$y = 4x^2 + \frac{1}{x^2} - 8,$$ find the rate of change of $x$ when the $y$-coordinate decreases at 5 units per second and $x=2$. 2. Different

1. Stating the problem: Differentiate the function $$f(x) = \frac{2}{x^2 - 3x + 1}$$ with respect to $$x$$. 2. Rewrite the function as $$f(x) = 2(x^2 - 3x + 1)^{-1}$$ to apply the

1. First, state the problem: Differentiate the function $$f(x) = \frac{2}{x^2 - 3x + 1}$$. 2. Rewrite the function as $$f(x) = 2 (x^2 - 3x + 1)^{-1}$$ to make differentiation easie

1. Énonçons le problème : calculer la dérivée d'une fonction donnée. 2. Soit une fonction $f(x)$, la dérivée $f'(x)$ est définie comme la limite de :

1. Problem 2.2.1: Find $\frac{dy}{dx}$ if $y = \cos\left(\frac{1+x^2}{1-x^2}\right)$. Step 1: Let $u = \frac{1+x^2}{1-x^2}$.

1. Given problem 2.2.1: Find $\frac{dy}{dx}$ for $y = \cos\left(\frac{x - 1}{1 + x^2}\right)$.\n\n2. Use the chain rule: $\frac{dy}{dx} = -\sin\left(\frac{x - 1}{1 + x^2}\right) \c

1. The problem is to find the limit of a function as the variable approaches a certain value. 2. Please provide the expression or function for which you want to calculate the limit

1. **Problem Statement:** Given the function $f(x) = \frac{1}{\sqrt{3x^3}}$, we want to find: 2.1.1 $f(x+h)$

1. We are given the function $$f(x) = \frac{1}{\sqrt[3]{3x}} = \frac{1}{(3x)^{1/3}}$$ and asked to find expressions related to increments in $x$. 2. Calculate $$f(x+h)$$ by substit

1. **State the problem:** Given the function $$f(x) = \frac{1}{\sqrt{3x^3}}$$, find expressions for: 2.1.1 $$f(x+h)$$

1. Problem: Find the derivative $f'(x)$ of the function $$f(x) = \pi \tan(x) e^x x^4.$$ 2. Write $f(x)$ as a product of three functions: $$u = \pi \tan(x), \quad v = e^x, \quad w =