∫ calculus

1. **State the problem:** We need to find the 2025th derivative of the function $g(x) = \cos(x)$. 2. **Recall the derivatives of cosine:**

1. **Stating the problem:** We are given that $y=f(x^2)$ and the derivative $\frac{dy}{dx} = 2x^3 - 4x$. We need to find $f'(4)$. 2. Since $y = f(x^2)$, by the chain rule,

1. **State the problem:** Find the derivative of the function $$y = (x^3 + 5)^x$$. 2. **Understand the function:** This is a function where the base and the exponent both depend on

1. **State the problem:** Find the derivative of the function $$y = \frac{x+1}{x-1}$$ with respect to the variable $$t = \sqrt{2x+1}$$ at $$x = 4$$. 2. **Explain the relation:** We

1. The problem is to find the derivative of the inverse function $g(x)$ of $f(x) = 5x + 2$ at $x=12$. 2. Recall that if $g$ is the inverse of $f$, then $g(f(x))=x$ and also $g'(y)

1. We are given two functions: $f(x) = \tan(x)$ and $g(x) = \frac{\pi}{x}$. We need to find the derivative of $f$ evaluated at $g(4)$, which is $f'(g(4))$. 2. First, evaluate $g(4)

1. **State the problem:** We need to find the derivative of the composition function $(f \circ g)(x) = f(g(x))$ evaluated at $x = \sqrt{\pi}$ given $f(x) = \sin(x)$ and $g(x) = x^{

1. The problem states: Given the function $f(x) = \sec^2 x - \tan^2 x$, find the derivative evaluated at $x = -1$, i.e., calculate $f'(-1)$.\n\n2. Recall the trigonometric identity

1. State the problem: Given $$y = \frac{1 + \tan x}{1 - \tan x}$$, find $$\frac{dy}{dx}$$ and identify it from the options. 2. Recognize that $$y = \frac{1 + \tan x}{1 - \tan x}$$

1. Given the function $$f(x) = ax^3 + 3x^2 + 4x + 1$$ and the condition $$f''(1) = 6$$. 2. We need to find the value of $$a$$.

1. **State the problem:** Given the function $f(x) = x^3 - 5x^2 + 9x - 3$, we need to find the value of the second derivative at $x=0$, i.e., find $f''(0)$. 2. **Find the first der

1. **State the problem**: Given the parametric equations $$ x = \sqrt{3t - 2} $$ and $$ y = \sqrt{4t + 1} $$, find the derivative $$ \frac{dy}{dx} $$ at $$ t = 2 $$. 2. **Different

1. **State the problem:** We are given two functions of variable $n$: $$y = n^3 - 1$$

1. **Stating the problem:** We are given the implicit equation $$x^3 + y^2 - 7x + 5y = 8$$ and we want to find $$\frac{dy}{dx}$$. 2. **Differentiate both sides implicitly with resp

1. We start with the given equation: $$3^y = \sin x + \cos x$$. 2. To find $\frac{dy}{dx}$, differentiate both sides with respect to $x$ using implicit differentiation.

1. Problem: Given the implicit relation $x^2 y^3 = 8$, find $\frac{dy}{dx}$ at $x = -1$. 2. Differentiate implicitly with respect to $x$:

1. Given the implicit function $x^2 y^3 = 8$, find $\frac{dy}{dx}$ at $x = -1$. Differentiate both sides implicitly using the product and chain rules:

1. **State the problem:** Find the equation of the normal to the curve $y = x^3 - 3x^2 + 2x$ at the point $P(3,6)$. 2. **Find the derivative:** The slope of the tangent to the curv

1. Solve the inequality $$\sqrt{x^2 - 3} \leq \frac{2}{\sqrt{x} - 2}.$$ Step 1: Determine the domain. Inside the square root, $x^2 - 3 \geq 0 \Rightarrow |x| \geq \sqrt{3}$. Also,

1. First, restate the problem: Differentiate the equation $$e^{x+y} - 3xy - 2 = y$$ with respect to $$x$$. 2. We will use implicit differentiation because $$y$$ is a function of $$

1. The problem is to differentiate the given equation with respect to $x$. 2. Since the equation was not explicitly provided, let's assume you want to differentiate a general funct