Calculus 3: Partial derivatives

Consider the function f(x,y) = xy^2 + x^3, and find the partial derivatives with respect to x and y.

What is the difference between a partial derivative and a total derivative of a function $f(x, y)$ when differentiated with respect to x?

Compute the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for the function $f(x, y) = x^2 \cdot y + \sin(y)$.

If the temperature distribution over a flat slab of metal is described by a function of two variables, like $f(x, y) = 9 - x^2 - y^2$, what is the partial derivative of this function with respect to $x$ and $y$?

Find the partial derivative of $z$ with respect to $x$ and the partial derivative of $z$ with respect to $y$ using the implicit function theorem for the equation $x^2 + y^4 - z^3 + 3xy^2 - 8 = 0$.

Using the implicit function theorem, find the partial derivative of $z$ with respect to $x$ and $y$ for the equation $xy^3 + x^2z^2 - 6 = 0$.

Find the derivative of $(\ln x)^7$.

Find the derivative of $\frac{1}{(x^2 + 8)^3}$.

Find $\frac{dW}{dT}$ for $W = x \cdot \sin(y)$ where $x = e^t$ and $y = \pi - t$, and evaluate $\frac{dW}{dT}$ at $t = 0$.

Calculate the partial derivative of a function Z with respect to X, holding Y constant.

Calculate the partial derivative of a function Z with respect to Y, holding X constant.

Given a multivariable function $f(x,y) = x^2 + y^2$, find where the partial derivatives are equal to zero to identify candidates for maximums or minimums.

The dimensions of a rectangular box are 75 cm, 60 cm, and 40 cm, with each measurement correct to within 0.2 cm. Use differentials to estimate the largest possible error when the volume of the box is calculated.