∫ calculus

1. The problem is to simplify the expression for the derivative: $$f'(x) = - \frac{(1+x)^2}{1-x} \cdot \frac{1+x}{1+x}$$

1. The problem asks to find the value of $c$ in Lagrange's Mean Value Theorem (MVT) for the function $f(x) = x(x - 1)$ on the interval $[1, 2]$. 2. Recall that Lagrange's MVT state

1. The problem is to find the derivative of the natural logarithm of a function $f(x)$ with respect to $x$. 2. Recall the chain rule for derivatives: if $y = \ln(f(x))$, then the d

1. Problem 108: Find $f'(x)$ for $f(x) = (x^2 + 1)(x^3 + 3)$ in two ways. (a) Multiply first, then differentiate:

1. **State the problem:** We are given the function $$f(x) = -2\sqrt{x^3} - \sqrt{x}$$ and need to find its derivative at $$x=3$$, i.e., $$f'(3)$$. 2. **Rewrite the function using

1. **State the problem:** We are given the function $$f(x) = \frac{5\sqrt{x}}{3} + 2\sqrt{x^3}$$ and need to find its derivative at $$x=1$$, i.e., $$f'(1)$$. We will express the an

1. **State the problem:** We are given the function $$f(x) = \frac{5\sqrt{x}}{3} + 2\sqrt{x^3}$$ and need to find its derivative at $$x=1$$, i.e., $$f'(1)$$. We will express the an

1. **State the problem:** We are given the function $$f(x) = \frac{2}{x} - \frac{1}{2x^2}$$ and need to find the derivative at $$x=2$$, i.e., $$f'(2)$$. 2. **Rewrite the function:*

1. **State the problem:** We are given the function $$f(x) = -\frac{2 \sqrt{x}}{5} + \frac{2 \sqrt{x^3}}{3}$$ and need to find its derivative at $$x=4$$, i.e., $$f'(4)$$. 2. **Rewr

1. **State the problem:** Given the function $$f(x) = \frac{5}{4\sqrt{x}} - \frac{\sqrt{x^3}}{4}$$, find the derivative $$f'(x)$$ and then evaluate $$f'(1)$$. Express the answer as

1. Evaluate $$\lim_{x \to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x - 2}$$. Step 1: Recognize this limit is of the form $$\frac{f(x) - f(2)}{x - 2}$$ where $$f(x) = \frac{1}{x}$$.

1. **State the problem:** We are given the function $$f(x) = \frac{5}{4\sqrt{x}} - \frac{3\sqrt{x^3}}{4}$$ and need to find its derivative at $$x=1$$, i.e., $$f'(1)$$. The answer s

1. **State the problem:** We are given the function $$f(x) = \frac{1}{\sqrt{x}} + \frac{3\sqrt{x}}{2}$$ and need to find its derivative at $$x=6$$, i.e., $$f'(6)$$, expressed as a

1. **State the problem:** We are given the function $$f(x) = - \frac{4}{3\sqrt{x}} - 2\sqrt{x}$$ and need to find its derivative at $$x=2$$, expressed as a single fraction in simpl

1. Problem: Find the limit $$\lim_{x \to -7} (2x + 5)$$ Step 1: Substitute $x = -7$ directly since the function is linear and continuous.

1. The problem asks us to find the derivative of the function $$f(x) = -\frac{1}{2x^3}$$ and express the answer using negative exponents. 2. First, rewrite the function using negat

1. The problem is to find the derivative of the function $$f(x) = -\frac{4\sqrt{x}}{3}$$. 2. First, rewrite the square root in exponent form: $$\sqrt{x} = x^{\frac{1}{2}}$$.

1. **State the problem:** We are given the function $f(x) = \frac{4}{5x^5}$ and need to find its derivative $f'(x)$. 2. **Rewrite the function:** Express $f(x)$ with negative expon

1. Evaluate each limit using limit theorems: (i) $$\lim_{x \to 3} (2x + 4) = 2(3) + 4 = 6 + 4 = 10$$

1. **State the problem:** We want to show that the functions defined by the series

1. The user provided several limit expressions and functions to analyze. 2. Let's clarify and solve each limit step-by-step.