∫ calculus

1. **Stel het probleem vast:** Bereken de dubbele integraal $$\int_0^1 \left( \int_{-y}^y e^{x+y} \, dx \right) dy$$. 2. **Formule en belangrijke regels:** De binnenste integraal i

1. **مشكلة:** إيجاد مشتقة الدالة $f(س) = س^3 - 2س + 5$. 2. **الصيغة:** مشتقة دالة كثيرة الحدود تُحسب باستخدام قاعدة القوة: $$\frac{d}{dس} س^n = n س^{n-1}$$

1. **อันดับและดีกรีของสมการเชิงอนุพันธ์** - อันดับของสมการเชิงอนุพันธ์คืออันดับของอนุพันธ์ที่สูงที่สุดในสมการ

1. **State the problem:** We want to find values of $a$ and $b$ such that for any linear function $g(x) = cx + d$, the integral $$\int_0^1 f(x)g(x) \, dx = 0$$ always holds. 2. **E

1. **Problem:** Find $\frac{dy}{dx}$ for $y = 3^{4x}$. **Step 1:** Use the exponential derivative formula $\frac{d}{dx} a^{u} = a^{u} \ln a \cdot \frac{du}{dx}$.

1. **Problem:** Find the derivative of $x^2\left(x-1\right)^{-1}$ with respect to $x$. 2. Use the product rule. If $f(x)g(x)$ is a product, then $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x

1. The problem is to understand and apply the quotient rule in calculus to differentiate a function that is the ratio of two functions. 2. The quotient rule formula is:

1. **Probleemstelling:** We willen bepalen hoe we kunnen weten of een gevonden oplossing een maximum of minimum is bij optimalisatieproblemen.

1. **Problem statement:** Find the percentage change in the area of a rectangle whose one side is four times the other, given that the perimeter decreases by 2%.

1. **State the problem:** We need to find the integral $$\int \frac{1}{1+x^2} \, dx$$. 2. **Recall the formula:** The integral of $$\frac{1}{1+x^2}$$ with respect to $$x$$ is a sta

1. **Probleemstelling:** We willen de extrema (maxima of minima) van een functie $$f(x_1, ..., x_n)$$ vinden, maar onder de randvoorwaarde dat $$g(x_1, ..., x_n) = 0$$ geldt.

1. **Stel het probleem vast:** We willen de extrema (maxima of minima) van een functie $f(x_1,...,x_m)$ vinden onder de voorwaarde dat $g(x_1,...,x_m)=0$. 2. **Belangrijk concept:*

1. **State the problem:** We need to find the integral $$\int Ce^x(x+1)\,dx$$ where $C$ is a constant. 2. **Recall the formula and rules:** The integral of a product involving $e^x

1. **Nyatakan masalah:** Diberikan turunan fungsi $f(x)$ yaitu $\frac{df(x)}{dx} = a + bx$ dan selisih nilai fungsi pada $x=1$ dan $x=0$ adalah $f(1) - f(0) = 3$. Kita diminta menc

1. The problem asks to find the first-order partial derivatives for multiple functions (a to i). Since the functions are not explicitly given, I will explain the general approach.

1. **State the problem:** Find the equation of the tangent line to the curve given by $$\frac{1}{3} y^3 + 2xy - 100 = 0$$ at the point $(-2, 2)$. 2. **Implicit differentiation:** D

1. **State the problem:** We have a spherical container with radius $r$ decreasing at a rate of $\frac{dr}{dt} = -2$ cm/min (since water is leaking, radius decreases).

1. **State the problem:** Evaluate the definite integral $$\int_3^6 \frac{x}{3\sqrt{x^2 - 8}} \, dx$$. 2. **Recall the formula and rules:** The integral involves a radical in the d

1. **State the problem:** We need to find the second partial derivative of $$z$$ with respect to $$x$$, where $$z = \frac{\sinh(x^2 y) e^{-y^2}}{2}$$. 2. **Recall the formula and r

1. **State the problem:** Find the slope of the tangent line to the inverse of the curve defined implicitly by $$x^2 = \cos^{-1}(xy)$$ at the point $$\left(\frac{1}{3}, 2\right)$$.

1. **State the problem:** We want to find the equation of the tangent line to the curve defined implicitly by $$\frac{1}{3} y^3 + 2xy - 100 = 0$$ at the point $(-2, 2)$. 2. **Recal