∫ calculus

1. Stating the problem: Find the derivative of the function $$f(x)=e^{3x} \sin(\ln(9+x))$$. 2. Use the product rule for differentiation: if $$f(x)=u(x)v(x)$$, then $$f'(x)=u'(x)v(x

1. **State the problem:** We need to find the derivative of the function $$f(x)=e^{\sin^3(7x)}$$. 2. **Rewrite the function:** The function can be written as $$f(x)=e^{(\sin(7x))^3

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{e^x}{x^3 + 2 \sin(x)}$$. 2. **Identify components:** This is a quotient of two functions:

1. We need to find the derivative of the function $$f(x)=x^5\cdot \sin(x)\cdot \cos(x)$$. 2. This is a product of three functions: $u=x^5$, $v=\sin(x)$, and $w=\cos(x)$. We will us

Problem 17: Find $\frac{dr}{d\theta}$ if $\sqrt{\theta} + \sqrt{r} = 1$. 1. Rewrite equation: $\theta^{1/2} + r^{1/2} = 1$

1. We are given the function derivative expression: $$f'(x) = \sqrt{\arccos(3x+15) - \arccos(x+25)}$$. 2. To understand or work with this expression, we note the domain restriction

**Problem:** Find $\frac{dy}{dx}$ for each pair given $y=f(u)$ and $u=g(x)$ using the chain rule $\frac{dy}{dx} = f'(g(x))g'(x)$ for Exercises 1 to 8. 1. Given $y=6u-9$ and $u=\fra

1. The problem asks to find the arc length of the given functions over the specified intervals. 2. Recall the arc length formula for a function $y=f(x)$ from $x=a$ to $x=b$:

1. **Problem Statement**: We are asked to find the values of A, B, C, D, and E in the integral expression

1. The problem asks to find the arc length of the curve defined by the function $y = x^3$ from $x=0$ to $x=2$. 2. The formula for the arc length $L$ of a function $y = f(x)$ from $

1. The problem is to find the indefinite integral $$\int (12x^6 + 7x^5 + 2)\,dx.$$\n\n2. We use the rule for integrating powers of $x$: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$

1. The problem asks us to find the indefinite integral of the function $12x^6 + 7x^5 + 2$ with respect to $x$. 2. We apply the power rule for integration to each term separately. R

1. Statement of the problem: Compute the integral $\int (12x^6+7x^5+2)\,dx$. 2. Reasoning: Use linearity to integrate term-by-term and the power rule $\int x^n\,dx=\frac{x^{n+1}}{n

1. **State the problem:** We need to evaluate the integral

1. Find the four second-order partial derivatives of the given functions. **For i.** $f(x,y) = 2x^2 y^3$

1. **Problem:** Evaluate $$\lim\limits_{(x,y)\to(0,\ln 2)} e^{x-y}$$ Step 1: Substitute $x=0$ and $y=\ln 2$ directly since exponential is continuous.

1. The problem is to find the limit $$\lim_{x\to 1}(5 - 4x)$$. 2. Recall that if the function is continuous at $x=1$, the limit can be found by direct substitution.

1. First, state the problem: Find the limit as $x$ approaches 1 of the function $$\frac{3x^3 + 2x^2 - 3x}{x^3 - 1}$$. 2. Substitute $x = 1$ directly into the expression to check if

1. Problem 6 asks to find the increasing and decreasing intervals for a function with two vertical asymptotes between $x=-2$ and $x=2$. The graph approaches $+\infty$ to the left o

1. **Problem statement:** Determine where the function $$f(x) = \frac{x - 1}{x^2 - 1}$$ is continuous and if possible, extend it to a larger domain continuously. 2. **Analyze the d

1. The problem asks to determine the continuity of a function, but no specific function was provided. 2. To analyze continuity, we first need the explicit mathematical expression o