∫ calculus

1. **State the problem:** We need to find the limit $$\lim_{h \to 0} \frac{\frac{5}{(2+h)^2} - \frac{5}{2^2}}{h}$$

1. **State the problem:** We need to find the derivative with respect to $x$ of the function $$f(x) = \ln(\sqrt{x}).$$ 2. **Rewrite the function:** Recall that $\sqrt{x} = x^{\frac

1. **State the problem:** We need to find the derivative of the function $$y=(1+5x-x^2)^{\frac{1}{4}}$$ and then evaluate it at $$x=5$$. 2. **Recall the formula:** For a function o

1. We are asked to find the derivative of the function $$y = \arcsin\left(\frac{x}{6}\right)$$ and then evaluate it at $$x=4$$. 2. Recall the derivative formula for $$y = \arcsin(u

1. We are given the implicit equation $$x^3 + y^2 = 24$$ and asked to find the second derivative $$\frac{d^2y}{dx^2}$$ at the point $$(2,4)$$. 2. First, differentiate both sides wi

1. **State the problem:** We need to find the second derivative with respect to $x$ of the function $$f(x) = e^{7x-4}$$. 2. **Recall the formula:** The derivative of an exponential

1. The problem is to find the derivative $\frac{dy}{dx}$ of a function at the point $(2,4)$. However, the function $y=f(x)$ is not provided, so we cannot directly compute the deriv

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{(\ln(x))^3}{x^3}.$$\n\n2. **Formula and rules:** We will use the quotient rule for

1. We are given the implicit equation $$3y^2 + x^2 - xy = 1$$ and asked to find $$\frac{dy}{dx}$$. 2. To find $$\frac{dy}{dx}$$ for implicit functions, we use implicit differentiat

1. **State the problem:** Simplify the expression and find the derivative where applicable: $2\sqrt{x}3 - 5 - \frac{d}{dx} 11 \sqrt{x} 12 - 1$. 2. **Rewrite the expression clearly:

1. **State the problem:** Given the function $$y = \frac{x+1}{\sqrt{x-1}}$$, show that $$2\sqrt{x-1} \left(\sqrt{x-1} \frac{dy}{dx} - 1\right) = -y = -\frac{x+1}{\sqrt{x-1}}$$. 2.

1. **State the problem:** Find the left-hand limit as $x$ approaches $-2$ of the expression $$\frac{1}{x-2} - \frac{1}{x^2-4}.$$\n\n2. **Recall the formula and factorization:** Not

1. The problem is to find the limit \( \lim_{x \to -2} \sqrt{x^2 + 4x + 4} \). 2. First, recognize that the expression inside the square root is a quadratic: \(x^2 + 4x + 4\).

1. **State the problem:** Find the limit as $x$ approaches $-\infty$ of the expression $$\frac{3x+5}{\sqrt{9x^2+2x+6}}.$$\n\n2. **Recall the formula and rules:** When dealing with

1. The problem is to find the integral with respect to $x$ of the function $5e^x - 2e^{4x^2}$. 2. Recall the integral rules:

1. **State the problem:** Find the limit $$\lim_{x \to -2} \frac{x^2 - 4}{x^3 + 8}$$. 2. **Recall the formulas and rules:**

1. **Problem:** Evaluate $$\lim_{x \to 2} \frac{x^2 + 3x + 2}{x^3 + 6x}$$. 2. **Formula and rules:** To find limits of rational functions as $x$ approaches a value, substitute the

1. **Problem statement:** Determine whether the improper integral $$\int_1^{\infty} \frac{\ln x}{x} \, dx$$ converges or diverges. 2. **Recall the integral test and behavior of fun

1. **State the problem:** Calculate the integral $$\int x \sin x \, dx$$. 2. **Formula and method:** Use integration by parts, which states:

1. The problem is to find the limit: $$\lim_{n \to \infty} \frac{1}{\ln n}$$. 2. Recall that the natural logarithm function $\ln n$ increases without bound as $n$ approaches infini

1. The problem is to find the derivative of the function $$f(x) = x \sqrt{4 - x}$$ and explain how to add fractions together into one fraction during the process. 2. First, recall