∫ calculus

1. The problem asks us to differentiate the function $y=\tan x$. 2. Recall that the derivative of $\tan x$ with respect to $x$ is a standard derivative result from calculus.

1. The problem is to find the integral of the function $\sec^2(x)$. 2. Recall that the derivative of $\tan(x)$ is $\sec^2(x)$, so the antiderivative (integral) of $\sec^2(x)$ is $\

1. The problem is to find the integral of $\tan^2 x$ with respect to $x$. 2. Recall that $\tan^2 x = \sec^2 x - 1$ from the Pythagorean identity.

1. We are asked to find the integral of the function $\tan x$. 2. Recall that $\tan x = \frac{\sin x}{\cos x}$.

1. Stating the problem: Find the derivatives of the functions \(y = 3x^{-4} + 2x^{-3} + x^{-2} + x^{-1}\) and \(y = \cos^3(2x)\). 2. For \(y = 3x^{-4} + 2x^{-3} + x^{-2} + x^{-1}\)

1. Diberikan fungsi $f(x) = 6 - 2x$.\nTurunan dari $f(x)$ adalah turunan dari setiap suku.\nTurunan dari konstanta 6 adalah 0, dan turunan dari $-2x$ adalah $-2$.\nJadi, $$f'(x) =

1. The problem is to find the derivatives of each given function. 2. For $y=\sqrt{x}$, rewrite as $y=x^{1/2}$. Using the power rule, $\frac{dy}{dx}=\frac{1}{2}x^{-1/2}=\frac{1}{2\s

1. **State the problem:** We have the function $f(x) = \sqrt{4 + \ln(x)}$. We want to find its derivative $f'(x)$ and determine its domain. 2. **Find the derivative:**

1. **Problem:** Find the derivative and domain of the function $$f(x) = \sqrt{4 + \ln(x)}$$ and analyze related domain conditions for $$g(x) = \ln(x^2 - 12x)$$. 2. **Derivative of*

1. Problem: Calculate \(\lim_{\theta \to 0} \frac{\tan 5\theta}{\sin 2\theta}\). Step 1: Using the standard limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \frac

1. The problem is to perform the division involving one integral expression. 2. Let's consider an example: divide the integral $$\int_0^1 x^2\,dx$$ by a number, say 2.

1. Let's start by stating the problem: You want to learn how to divide integrals. 2. Remember that integrals themselves cannot be directly divided like regular numbers or expressio

1. The problem is to integrate the expression $$x^2 + \frac{1}{6}x - 9x^2$$ with respect to $$x$$. 2. First, combine like terms: $$x^2 - 9x^2 = -8x^2$$, so the expression inside th

1. **State the problem:** Calculate the integral $$\int \left(x^2 + \frac{1}{6}x - 9x^2\right) \, dx$$ and explain the steps clearly. 2. **Simplify the integrand:** Combine like te

1. Stated problem: Differentiate $T(z) = 4^z \log_4(z)$. 2. Recall the product rule for derivatives: if $T(z) = u(z)v(z)$, then $$T'(z) = u'(z)v(z) + u(z)v'(z)$$

1. The problem is to find the derivative of the function $F(t) = (\ln(t))^2 \cos(t)$.\n\n2. We apply the product rule for differentiation: if $F(t) = u(t) v(t)$, then $F'(t) = u'(t

1. The minimum point of the graph of a function $f(x)$ is the point where the function reaches its lowest value in a particular region. 2. To find the coordinates of the minimum po

1. The problem asks for the one-sided limit of the function $f(x)$ as $x$ approaches 2 from the right, i.e., $\lim_{x\to 2^+} f(x)$. 2. According to the graph description, there is

1. We are asked to find the one-sided limit $$\lim_{x\to 2^+} f(x)$$ from the graph of the function $$f$$. 2. The graph indicates a vertical asymptote at $$x=2$$, shown by the dash

1. The problem asks for the one-sided limit \(\lim_{x\to 2^+} f(x)\), which means the value that \(f(x)\) approaches as \(x\) approaches 2 from the right side (values greater than

1. The problem asks for the value of $$\lim_{x\to -2} f(x)$$ based on the given graph of the function $$f$$. 2. To find the limit as $$x$$ approaches $$-2$$, we examine the $$y$$-v