∫ calculus

1. **Problem Statement:** Find the slope of the tangent line to the graph of $f(x) = \tan(x)$ at $x = \frac{\pi}{6}$. 2. **Formula:** The slope of the tangent line to a function at

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\ln(x)}{(x-1)^2}$$. 2. **Recall the formula:** To differentiate a quotient $$\frac{u}{v}$$, use the quo

1. **State the problem:** Find the limits of the function $$f(x) = \frac{\ln(x)}{(x-1)^2}$$ as $$x \to +\infty$$, $$x \to 0^+$$, $$x \to 1^+$$, and $$x \to 1^-$$. 2. **Recall impor

1. **State the problem:** We want to find the direction of change (increasing or decreasing) of the function $$f(x) = \sqrt{x} - 1 - \sqrt{x} \ln(x)$$ for $$x > 0$$ since the funct

1. **State the problem:** Find the derivative of the function $$f(x) = \sqrt{x} - 1 - \sqrt{x} \ln(x)$$. 2. **Recall the formulas and rules:**

1. **State the problem:** Find the limit of the function $$f(x) = \sqrt{x - 1} - \sqrt{x} \ln(x)$$ as $$x \to +\infty$$. 2. **Recall the behavior of components:**

1. The problem is to analyze the behavior of a function as $x$ approaches $+\infty$ (positive infinity). 2. When $x \to +\infty$, we look at the limit $\lim_{x \to +\infty} f(x)$ t

1. **State the problem:** We want to find the limit of the function $$f(x) = \sqrt{x} - 1 - \sqrt{x} \ln(x)$$ as $$x$$ approaches $$0^+$$ (from the right side). 2. **Recall the beh

1. **Problem:** Find the derivative $\frac{d}{dx}(\sec^2(x^3))$. 2. **Formula and rules:** Use the chain rule: if $y = [u(x)]^2$, then $\frac{dy}{dx} = 2u(x) \cdot u'(x)$. Also, $\

1. **State the problem:** Find the third derivative $\frac{d^3y}{dx^3}$ of the function $$y = e^{-x}(\cos 2x + \sin 2x).$$ 2. **Recall the product rule and chain rule:** For deriva

1. The problem is to identify which graphs represent the gradient (derivative) functions of given graphs A, B, and D, and to find one other pair of graphs where one is the gradient

1. **Problem:** Find the inverse of the logarithmic function $$y = \log_5(2x - 1) - 7$$. 2. **Recall the definition of inverse functions:** If $$y = f(x)$$, then the inverse functi

1. **Problem:** Sketch and analyze the function $f$ with the given concavity and monotonicity properties. 2. **Key concepts:**

1. The problem is to understand if you need to find the derivative every time to calculate the rate of change. 2. The rate of change of a function at a point is given by the deriva

1. **State the problem:** We are given the temperature function $$T = 5 \sin\left(\frac{\pi}{12} x\right) + 23$$ where $x$ is the number of hours after sunrise. We need to find the

1. The problem is to find the derivative of the function $y = x^{\cos^{-1} x}$. 2. We use the formula for the derivative of $y = f(x)^{g(x)}$:

1. **Problem Statement:** Find the second derivative of the function $$f(\theta) = \frac{1}{3 + 2\cos\theta}$$ with respect to $$\theta$$. 2. **Recall the formula:** To find the se

1. The problem asks for the value of the limit $$\lim_{x \to -4^+} f(x)$$, which means the limit of the function $f(x)$ as $x$ approaches $-4$ from the right side. 2. From the grap

1. **State the problem:** Find the limit $$\lim_{x \to 0} (1 + 2^x)^{\frac{3}{x}}.$$\n\n2. **Recall the formula:** Limits of the form $$\lim_{x \to 0} (1 + f(x))^{\frac{1}{x}} = e^

1. Problem: Find the derivative of $y = \sin \alpha x$. 2. Formula: The derivative of $\sin u$ with respect to $x$ is $\cos u \cdot \frac{du}{dx}$.

1. Problem: Find the derivative of the function $f(x) = \frac{2}{3} x^{-3}$.\n\n2. Recall the power rule for derivatives: If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$.\n\n3. Apply