∫ calculus

1. **State the problem:** Find the derivative of the function $$f(x) = 2^x \log_3 \left(7^{x^2 - 4}\right)$$. 2. **Rewrite the function:** Use the logarithm power rule $$\log_a b^c

1. The problem is to find the derivatives of the functions \(g(t) = \frac{1}{t^2}\), \(k(z) = \frac{1 - z}{2z}\), and \(p(\theta) = \sqrt{3\theta}\) using the definition of the der

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ of the function $$y = x^{2} \cos \left(\sqrt{x^{3} - 1} + 2 \right)$$

1. The topics you mentioned cover advanced calculus concepts including limits, continuity, differentiability, Rolle's theorem, Mean Value Theorems, Maclaurin and Taylor expansions,

1. **State the problem:** We want to find the integral of the function $$\frac{1}{3}(2-x) + \frac{1}{3}(x+1)$$ with respect to $x$. 2. **Simplify the integrand:** Combine the terms

1. **State the problem:** Differentiate the equation $$3(x-y)^2 = 2xy + 1$$ with respect to $x$. 2. **Recall the rules:** We will use implicit differentiation since $y$ is a functi

1. **State the problem:** We need to find the limit $$\lim_{x \to 1} (x^3 - 1)$$. 2. **Recall the formula:** The expression is a difference of cubes, which can be factored using th

1. **State the problem:** Find the derivative $f'(x)$ of the function $f(x) = 2x^2 + 3x + 4$. 2. **Recall the derivative rules:**

1. Statement of the problem: We are given the graph of a differentiable function f and the line tangent to the graph at x = 2; find $f'(2)$. 2. Formula and rules: The derivative at

1. **Problem statement:** We are given the graph of a differentiable function $f$ and a tangent line to the graph at $x=2$. We need to find the value of the derivative $f'(2)$, whi

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x}$$. 2. **Recall the formula and rules:** When evaluating limits involving square roots that re

1. **State the problem:** Find the global maximum and minimum values of the function $$f(x) = 2x^3 - 3x^2 - 12x + 1$$ on the interval $$[0, 3]$$. 2. **Formula and rules:** To find

1. The problem asks whether 0 is a local minimum or maximum of the function above. 2. To determine this, we need the function and its derivatives. Since the function is not provide

1. **Problem Statement:** Determine if the function $f(x) = x^3$ has a turning point. 2. **Formula and Rules:** A turning point occurs where the first derivative $f'(x)$ changes si

1. **State the problem:** Find the global maximum and minimum values of the function $$f(x) = 2x^2 - 8x + 7$$ on the interval $$[-3, 3]$$. 2. **Recall the formula and rules:** To f

1. The problem is to find the derivative of a function using $\frac{dy}{dx}$ notation. 2. The derivative $\frac{dy}{dx}$ represents the rate of change of $y$ with respect to $x$.

1. **Stating the problem:** Find the second derivative of the function $$y(t) = \frac{k}{1 + Ae^{-rt}}$$ with respect to $t$. 2. **Recall the formula:** The function is a quotient,

1. **State the problem:** We need to find the integral of the function $$\frac{x^3 + 5}{x^2 - 25}$$ with respect to $x$. 2. **Recall the formula and rules:** To integrate a rationa

1. **State the problem:** We need to find the integral of the function $$x^3 + \frac{5}{x^2} - 25$$ with respect to $$x$$. 2. **Rewrite the integral:** The integral is $$\int \left

1. **State the problem:** We need to find the area bounded by the curve $y = x^2 - 2x$, the x-axis, and the vertical lines (ordinates) $x = -2$ and $x = 3$. 2. **Formula and explan

1. **State the problem:** Show that the integral of $a^x$ with respect to $x$ is $$\int a^x \, dx = \frac{e^{x \ln a}}{\ln a} + C$$ where $a > 0$ and $a \neq 1$. 2. **Recall the fo