∫ calculus

1. Vous pouvez m'envoyer la photo de la fonction. 2. Je vais identifier la fonction mathématique à partir de l'image.

1. **State the problem:** Find the derivative of the function $$y=\frac{x^3}{x^2+1}$$. 2. **Formula used:** To differentiate a quotient, use the quotient rule:

1. **State the problem:** Find the derivative of the function $$y = x^2 (x-3)^2$$. 2. **Formula and rules:** We will use the product rule for derivatives, which states:

1. **State the problem:** We need to evaluate the integral $$\int (x + 2) \sqrt{x^2 + 10x - 11} \, dx.$$\n\n2. **Identify the integral type and substitution:** The integrand involv

1. **State the problem:** Find the derivative of the function $$y=\frac{\sqrt{x}}{(1-2x)^2}$$. 2. **Recall the formula:** We will use the quotient rule for derivatives, which state

1. **Stating the problem:** We need to find the implicit derivative $\frac{dy}{dx}$ of the equation $$e^x - x^e \over e^x - y = e^{\sqrt{xy}}.$$\n\n2. **Rewrite the equation:** The

1. **State the problem:** Find the implicit derivative $\frac{dy}{dx}$ of the equation $$e^x - x^e \over e^x - y = e^{\sqrt{xy} + 1}.$$\n\n2. **Rewrite the equation:** \n$$\frac{e^

1. **State the problem:** Find the implicit derivative $\frac{dy}{dx}$ of the equation $$\frac{e^x - x^e}{e^x - y} = e^{\sqrt{xy} + 1}.$$\n\n2. **Recall the rules:** We will use th

1. **State the problem:** Find the first and second derivatives of the function $$w = 3z^{-2} - \frac{1}{z}$$. 2. **Recall the power rule for derivatives:**

1. **Problem Statement:** Find the derivative of the function \( w = \left(1 + \frac{3z}{3z}\right)(3 - z) \) using the product rule. 2. **Simplify the function first:** Note that

1. Problem: Find the derivative of $y = x^{12}(1 + x^2)$ using the product rule. 2. Formula: The product rule states that if $y = uv$, then $$\frac{dy}{dx} = u \frac{dv}{dx} + v \f

1. **Problem:** Find the derivative of $$y = \frac{x^5 + 57x^2}{5x^3 - 6x}$$ using the quotient rule. 2. **Formula:** The quotient rule states:

1. **Problem:** Find the derivative of $$\frac{x^5 + 57x^2}{5x^3 - 6x}$$. 2. **Formula:** Use the quotient rule: $$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\f

1. The problem is to find the derivative of the function $$y = \frac{7}{x}$$. 2. Rewrite the function using exponents: $$y = 7x^{-1}$$.

1. **State the problem:** Given the function $y = \frac{7}{x}$, find the derivative $\frac{dy}{dx}$. 2. **Recall the formula:** The derivative of $y = x^n$ is $\frac{dy}{dx} = nx^{

1. **Problem Statement:** Given the graph and points, find the values of $f(-2)$, $\lim_{x \to 0^-} f(x)$, $\lim_{x \to 0^+} f(x)$, determine if $\lim_{x \to 0} f(x)$ exists, and f

1. **Problem:** Determine the convergence or divergence of the series \(\sum \frac{1}{n \ln(n)}\) using the Comparison Test. **Step 1:** For all \(n > 1\), note that \(\frac{1}{n \

1. The problem involves determining the convergence or divergence of series using the Comparison Test. 2. The Comparison Test states: If $0 \leq a_n \leq b_n$ for all $n$ beyond so

1. **Problem:** Find the limit \(\lim_{x \to 5} 8x - 2\). 2. **Formula and rule:** For limits of polynomial or linear functions, the limit as \(x\) approaches a value is simply the

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{5}{2x + 3}$$ using the alternative formula for derivatives: $$f'(x) = \lim_{z \to x} \frac{f(z) - f(x)}

1. **Problem statement:** Find the derivative of the function $w = z + \sqrt{z}$ and evaluate it at $z = 4$. 2. **Formula and rules:** The derivative of $w$ with respect to $z$ is