∫ calculus

1. **Stating the problem:** We need to understand and evaluate limits in calculus. 2. **Formula and rules:** The limit of a function $f(x)$ as $x$ approaches a value $a$ is written

1. **Stel het probleem vast:** We onderzoeken de lokale maxima en minima van de functie $$f: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto 2x^3 + 9xy^2 + 15x^2 + 27y^2$$. 2. **Zoek d

1. **Problem:** Find the antiderivative of $f(x) = \frac{1}{4} - \frac{2}{5}x^3$. 2. **Formula:** The antiderivative (indefinite integral) of $x^n$ is $\int x^n dx = \frac{x^{n+1}}

1. **Probleemstelling:** We willen de gradient van een functie $f(x,y)$ in $\mathbb{R}^2$ berekenen. 2. **Definitie van de gradient:** De gradient van een functie $f(x,y)$ is een v

1. Het probleem is om de formule te vinden die volgt na de tweede afgeleide. 2. De tweede afgeleide van een functie $f(x)$ wordt genoteerd als $$f''(x) = \frac{d^2}{dx^2} f(x)$$.

1. We beginnen met het begrijpen van wat de tweede afgeleide is. 2. De eerste afgeleide van een functie $f(x)$, genoteerd als $f'(x)$, geeft de helling of de verandering van $f(x)$

1. **Probleemstelling:** We willen de eerste en tweede afgeleide van $f$ bij $x=1$ vinden, gegeven de impliciete functie $F(x,y) = xy^3 - 2xy^2 + 3x^2y + 6x^3 - 6$ en de voorwaarde

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} (3x^2 + 7)$$. 2. **Recall the limit rule:** The limit of a sum is the sum of the limits, and the limit of a con

1. **State the problem:** We want to find the limit $$\lim_{x \to 1} \frac{\ln x}{x-1}$$. 2. **Recall the formula and important rule:** This is a classic limit that can be evaluate

1. **State the problem:** We need to evaluate the integral $$I = \int_{-2}^{1} |x^2 - x - 2| \, dx.$$ This integral involves the absolute value of a quadratic expression. 2. **Anal

1. Let's start by stating the problem: We want to understand what a limit is in mathematics. 2. A limit describes the value that a function $f(x)$ approaches as the input $x$ appro

1. **State the problem:** Find $\frac{dx}{dy}$ given the implicit equation $$x = \sqrt{xy} + 3$$.

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ using implicit differentiation for an equation involving both $x$ and $y$. 2. **Formula and Rules:** When differentiat

1. Differentiate $$y^3 = 2xy^2 + x^2y$$ with respect to $$x$$. Use implicit differentiation and product rule.

1. **State the problem:** Differentiate the equation $$\frac{y}{x} = \sqrt{xy} + 3$$ with respect to $$y$$. 2. **Rewrite the equation:**

1. **State the problem:** We are given the function $f(x) = 3e^x + 10 \ln(x)$ and need to find the derivative evaluated at $x=3$, i.e., $f'(3)$. 2. **Recall the derivative rules:**

1. **State the problem:** Find the derivative of the function $$y = x^2$$ at the point $$x = 3$$ using first principles.

1. **State the problem:** Calculate the value of

1. **State the problem:** Differentiate the equation $$\frac{y}{x} = \sqrt{xy} + 3$$ with respect to $$y$$. 2. **Rewrite the equation:**

1. **Problem:** Differentiate implicitly with respect to $x$ the equation $$y^3 = 2xy^2 + x^2y.$$ 2. **Formula and rules:** Use implicit differentiation. Recall the product rule: $

1. **Problem:** Evaluate $\int \sin 2\theta (\sin^2 \theta + 5)^4 d\theta$. **Step 1:** Recognize the chain rule pattern. Let $u = \sin^2 \theta + 5$.