∫ calculus

1. We begin by stating the problem: we need to find the derivative of the function $f(x)$ with respect to $x$, denoted $f'(x)$. 2. Since the function $f(x)$ is not explicitly given

1. The problem is to find the indefinite integral $\int \sin 2x \, dx$. 2. Recall the integral formula: $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C$ where $a$ is a constant.

1. Stating the problem: Given $y = 3^{2 \log_3 x}$, find the second derivative $\frac{d^2 y}{dx^2}$.\n\n2. Simplify the expression using properties of logarithms and exponents. Rec

1. We are asked to evaluate the limit as $n \to 0$ of the expression $$\int_0^\pi (\sin \theta - \sin \theta) \lim_{n\to 0} \frac{\sqrt{n+\theta} - \sqrt{n+\theta}}{\theta} d\theta

1. Problem 9: Find the functions $f(x)$ given their derivatives $f'(x)$. 2. To find $f(x)$, integrate each derivative function with respect to $x$. Remember to add a constant of in

1. **Problem Statement:** Given the function $$g(x) = \frac{x^2 + 9}{|x - 3|}$$ (a) Determine the domain of $$g(x)$$.

1. **State the problem:** Determine if the limit exists:

1. The problem is to compute the expression $l[e^{-2t}\sinh 4t]$. 2. Recall that the Laplace transform of $\sinh(at)$ is $\frac{a}{s^2 - a^2}$ for $s > |a|$.

1. We evaluate each limit step-by-step. 2. a) $$\lim_{x\to4} \frac{x^2 - 16}{x - 4}$$ Factor numerator: $$x^2 - 16 = (x-4)(x+4)$$ Cancel $$x-4$$ term:

1. The problem is to find the indefinite integral of the polynomial function $3x^2 + 7x - 2$ with respect to $x$. 2. We apply the power rule for integration: for each term $ax^n$,

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