∫ calculus

1. **Problem:** Given $x = \frac{1 - t^2}{1 + t^2}$ and $y = \frac{2t}{1 + t^2}$, prove that $$\frac{dy}{dx} + \frac{x}{y} = 0.$$\n\n2. **Find derivatives $\frac{dx}{dt}$ and $\fra

1. **Find the derivatives:** 1.1. Find $D_x (\ln 3 \sqrt[3]{x})$.

1. **State the problem:** Find the equation of the tangent line to the curve $$y = x^3 - 6x^2 + 9x + 4$$ at the point where $$x = 2$$. 2. **Find the derivative:** The derivative $$

1. Problem (iii): Evaluate $$\int (\sqrt{x} + \frac{1}{\sqrt{x}})^2 \, dx$$ Step 1: Expand the integrand:

1. The problem is to evaluate the definite integral $$\int_0^1 x \ln(x+1) \, dx.$$\n\n2. Use integration by parts. Let \(u = \ln(x+1)\) and \(dv = x \, dx\). Then \(du = \frac{1}{x

1. **بيان المسألة:** لدينا دالتان:

1. The problem asks us to explain why Rolle's theorem does not apply to the function $f(x) = \frac{1}{x} - 3$ on the interval $[-4,4]$. 2. Rolle's theorem states that if a function

1. **Problem a:** Find $$\lim_{x \to 0} \frac{\sin x}{x}$$ using Bernoulli-l’Hôpital’s rule. Since direct substitution gives $$\frac{0}{0}$$, apply l’Hôpital’s rule:

1. The problem is to find the derivative of the function $f(x) = e^{2 - x^2}$.\n\n2. Recall the chain rule for derivatives: if $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)

1. **State the problem:** We need to find the limit $$\lim_{x \to a} 2 \sin x$$. 2. **Recall the limit property:** The sine function is continuous everywhere, so $$\lim_{x \to a} \

1. **Problem a:** Find the derivative of $f(x) = (\tan x)^x$ with domain $D_f = ]0, \frac{\pi}{2}[$ using logarithmic differentiation. 2. Take the natural logarithm of both sides:

1. **State the problem:** We need to evaluate the definite integral $$\int_0^{\frac{\pi}{6}} x \sin(2x) \, dx$$. 2. **Use integration by parts:** Let $$u = x$$ and $$dv = \sin(2x)

1. **State the problem:** We want to evaluate the integral $$\int_1^{\infty} (4 + 2x + 6x^2) e^{-(5 + 4x + x^2 + 2x^3)} \, dx.$$\n\n2. **Analyze the integrand:** The integrand is a

1. The problem is to evaluate the definite integral $$\int_1^e \frac{1}{x^3} \, dx$$. 2. Rewrite the integrand using a negative exponent: $$\frac{1}{x^3} = x^{-3}$$.

1. The problem is to evaluate the definite integral $$\int_e^1 \frac{1}{x^3} \, dx$$. 2. Rewrite the integrand as $$x^{-3}$$ to make integration easier.

1. **Problem a:** Find local extrema, global max, and min of $f(x) = x^4 - 3x^3 + x^2 - 5$ on $[-5,5]$. 2. Compute derivative:

1. **State the problem:** Find the $n^{th}$ order derivative of the function $$y = \sin(5x) \cdot \cos(3x).$$ 2. **Use product rule:** Since $y$ is a product of two functions, $u =

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $$y = 4x^{8} + 4 \sqrt{x} - 5 + \frac{3}{x^{\frac{8}{5}}}.$$\n\n2. **Rewrite the function f

1. **State the problem:** Find the $n$th derivative of the function $$f(x) = \frac{1}{(x-1)(x-2)(x-3)}.$$\n\n2. **Rewrite the function:** We have $$f(x) = \frac{1}{(x-1)(x-2)(x-3)}

1. **State the problem:** We want to find the limit as $x$ approaches 0 of the expression $$\frac{a - \sqrt{a^2 - x^2}}{x}.$$ 2. **Understand the expression:** The numerator is $a

1. **Problem statement:** Find all local extrema, the global maximum, and the global minimum of the functions: c) $$f(x) = e^{-\frac{x^2}{2}}$$