∫ calculus

1. **State the problem:** We are given the function $$f(x,y) = e^{2x} \cos y$$ and need to find the partial derivatives $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\p

1. **State the problem:** We have a function $f(x,y) = x^2 y$ where $x = e^t$ and $y = \ln t$ for $t > 0$. We want to find $\frac{df}{dt}$ using the chain rule and express the answ

1. **State the problem:** We need to find the derivative $y'$ of the function $$y = (\sin x)^{\cos x}.$$\n\n2. **Rewrite the function using logarithms:** To differentiate a functio

1. **State the problem:** We are given the function $$w = x^2 y + y^2$$ where $$x = e^{5t}$$ and $$y = \sin t$$. We need to find $$\frac{dw}{dt}$$ using the chain rule and simplify

1. **Problem:** Find the tangent line to the function $f(x) = x^2 - 4x + 1$ at the point where $x=0$. Step 1: Calculate $f(0)$.

1. The problem is to find the integral of the function $x + 4$ with respect to $x$. 2. Recall that the integral of a sum is the sum of the integrals: $$\int (x + 4) \, dx = \int x

1. The problem is to find the integral of the function $x + 4$ with respect to $x$. 2. Recall that the integral of a sum is the sum of the integrals: $$\int (x + 4) \, dx = \int x

1. The problem is to evaluate the integral $\int x \, dx$. 2. Recall the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

1. The problem is to find the derivatives (Ableitung) of the two given functions: - $f(x) = 4x^3 - 2x + 1$

1. **Problem:** Show that the function $$f(x) = \frac{1}{x} \sin \frac{1}{x}$$ for $$x \neq 0$$ and $$f(0) = 0$$ is finite on $$[-1,1]$$ but not bounded, and determine any points o

1. **Problem statement:** We analyze the functions and limits given, solve equations, and study derivatives. 2. For $g(x) = x^3 + 3x + 8$, it has a unique root $\alpha$ in $(-2,0)$

1. The problem is to find the derivative of the function $f(x) = x^3$ with respect to $x$. 2. Recall the power rule for differentiation: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

1. The problem appears to be finding the derivative of the function $f(x) = x^3$. 2. To find the derivative $\frac{d}{dx}(x^3)$, we use the power rule of differentiation which stat

1. Differentiate $y = (4x)^2 - \sqrt{(4x)^2}$. Step 1: Simplify the expression.

1. Let's clarify the problem: finding the asymptote of the top left corner of a graph usually means identifying the behavior of the function as $x \to -\infty$ or near a vertical b

1. **State the problem:** Find the equation of the normal line to the curve $$y=\frac{x^3}{3}+x^2+3x+2$$ at the point (0,0). 2. **Verify the point lies on the curve:** Substitute $

1. **State the problem:** Find the volume of the sphere given by the equation $$x^2 + y^2 + z^2 = 16$$ using triple integration. 2. **Identify the radius:** The equation of the sph

1. **State the problem:** Find the volume of the sphere defined by the equation $$x^2 + y^2 + z^2 = a^2$$ using triple integration. 2. **Set up the integral:** The sphere is symmet

1. The problem appears to involve the expression \(\sin 7 \cos 3 x x dx\), which is ambiguous. Assuming you want to integrate the function \(f(x) = \sin(7) \cos(3x) x\) with respec

1. The problem is to evaluate the integral $$\int \frac{2x+1}{4x^2+13} \, dx$$. 2. Notice that the denominator is a quadratic expression $4x^2 + 13$ which cannot be factored easily

1. **State the problem:** We want to evaluate the integral $$\int \sqrt{x^3 - 3} (x^2 - 1) \, dx.$$\n\n2. **Substitution:** Let $$u = x^3 - 3.$$ Then, differentiate both sides with