∫ calculus

1. Given the function $$y = 8(5x^2 - 1)(x^2 - 1)$$, we want to find the second derivative $$y''$$. 2. First, expand the function:

1. Stating the problem: Find the third derivative $\frac{d^3y}{dx^3}$ of the function $$y=\frac{x^2+2x}{x-1}.$$\n\n2. Simplify the function first by performing polynomial division:

1. The problem: Find the third derivative ($y'''$) of the function $$y = \frac{x^2 + 2x}{x - 1}.$$\n\n2. To differentiate, let's rewrite $y$ as a quotient and apply the quotient ru

1. The problem is to find the third derivative, $y'''$, of the function $$y = \frac{x-1}{x^2 + 2x}.$$\n\n2. First, simplify the denominator: $$x^2 + 2x = x(x+2).$$\n\n3. Use the qu

1. **State the problem:** We need to find the third derivative $y'''$ of the function $$y=\frac{x^2+2x}{x-1}.$$\n\n2. **Rewrite the function to simplify differentiation:** Use the

1. State the problem: Find the third derivative $y'''$ of the function $y = x^2 - \sqrt[3]{4x}$. 2. Rewrite the function using exponents: $$y = x^2 - (4x)^{1/3}.$$

1. Énoncé du problème : Nous devons calculer $dF$ où $F = \sin(x) y^2$ et $\varphi(t) = (t^2, t)$. Ici, $F$ est une fonction de deux variables $x$ et $y$, et $\varphi$ est une fonc

1. We are asked to evaluate the integral $$\int 4x^2 \sqrt{x^3 - 5} \, dx$$. 2. Let us use substitution. Set $$u = x^3 - 5$$.