∫ calculus

1. Diberikan fungsi $f(x) = 6x^3 + 12x^2 + 5x + 2$. Kita diminta mencari turunan ketiga dari fungsi ini. 2. Ingat bahwa turunan fungsi polinomial dapat dihitung dengan aturan pangk

1. **State the problem:** We need to find the limit $$\lim_{x \to -\infty} \left(x - 1 + 2\sqrt{1 - x}\right).$$\n\n2. **Recall the behavior of terms as $x \to -\infty$:**\n- The t

1. Problem: Given the function $f(x) = (2x + 5)^8$, find the second derivative $f''(x)$ and determine the smallest $n$ such that the $n$th derivative $f^{(n)}(x)$ is a constant. St

1. The problem is to find the derivative with respect to $x$ of the expression $\frac{M_{Ed,x}}{z_x}$ and show that it equals zero. 2. The expression is $$\frac{d}{dx} \left( \frac

1. **State the problem:** Find the equation of the tangent line to the function $f(x) = e^{3x - 3}$ at the point where $x = 1$. 2. **Recall the formula for the tangent line:** The

1. **State the problem:** We need to find the first derivative of the function $$h(x) = \ln \left( \frac{x^2 - x}{x^3 - 1} \right).$$ 2. **Recall the derivative rule for logarithms

1. **Problem 1: Find the turning points of** $y = x^3 - 3x + 5$ and distinguish between them. 2. To find turning points, we first find the derivative $y' = \frac{dy}{dx}$ and set i

1. **State the problem:** We need to find $\frac{dy}{dx}$ using implicit differentiation for the equation $$x - \cos(x^2) + \frac{y^2}{x} + 3x^5 = 4x^3.$$\n\n2. **Recall the rules:

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicitly differentiating the equation $$x - \cos(x^2) + \frac{y^2}{x} + 3x^5 = 4x^3.$$\n\n2. **Recall the rules:**\n-

1. **State the problem:** We are given the function $$y = 7x - \sin^2(x)$$ and need to find its differential $$dy$$. 2. **Recall the formula for differential:** The differential of

1. **State the problem:** Find the differential $dy$ of the function $$y = \frac{x + 2}{2x - 3}.$$\n\n2. **Recall the formula:** For a function $y = \frac{u}{v}$, the differential

1. **State the problem:** Find the differential $dy$ of the function $y = \csc(3x)$.\n\n2. **Recall the formula:** The derivative of $\csc(u)$ with respect to $x$ is $\frac{d}{dx}[

1. **State the problem:** We are given the function $y = 7x^2 - 4$ and need to find its differential $dy$. 2. **Recall the formula:** The differential $dy$ of a function $y = f(x)$

1. **Problem 1: Find the turning (stationary) points of** $y = x^3 - 3x + 5$ **and distinguish between them.** 2. The turning points occur where the first derivative $y'$ is zero.

1. **Problem Statement:** Find the turning points and their nature for the curve $$y = \frac{x^3}{3} - \frac{x^2}{2} - 6x + \frac{5}{3}$$. 2. **Step 1: Find the first derivative $$

1. נתחיל בהגדרת הבעיה: נדרש למצוא את הגבול של הפונקציה $f(x) = \sin(\pi x)$ כאשר $x$ שואף ל-1 מימין, כלומר $\lim_{x \to 1^+} \sin(\pi x)$.\n\n2. נזכיר כלל חשוב לגבולות של פונקציות

1. **State the problem:** We are asked to find the limit of the piecewise function $$f(x) = \begin{cases} \sin(\pi x) & x > 1 \\ (x-1)^3 & x < 1 \end{cases}$$

1. لنفهم السؤال، لدينا دالة $f(x)$ وقيمة $f(x)$ أكبر من الصفر. 2. المطلوب هو معرفة إذا كان يمكن تعويض $f(x)$ عند نهاية الدالة عندما تقترب $x$ من الصفر من اليمين.

1. **Problem:** Differentiate $y = x^2$. **Formula:** Power rule: $\frac{d}{dx} x^n = n x^{n-1}$.

1. **State the problem:** Solve the integral $$\int \frac{x^2}{\sqrt{16 - x^2}} \, dx$$ using the trigonometric substitution $$x = 4 \sin \theta$$. 2. **Recall the substitution and

1. **Problem Statement:** Solve the integral $$\int \frac{\sqrt{x^2 - 36}}{x} \, dx$$ using the trigonometric substitution $$x = 6 \sec \theta$$. 2. **Formula and Substitution:** W