Calculus 3: Linearization, chain rule, gradient

Consider the function f(x,y,z) = $\frac{(x^5)(e^{2z})}{y}$, and find the gradient of the function.

Given a function $f(x, y) = x^2 \cdot y$, where $x(t) = 2t + 1$ and $y(t) = t^3$, find $\frac{dw}{dt}$ using the multi-variable chain rule.

What is the derivative of the function composition $F(x(T), y(T))$ given $F(x, y) = x^2 y$, $x(T) = \,\cos(T)$, and $y(T) = s(T)$?

Find the derivative of $(5x + 3)^4$.

Find the derivative of $(x^2 - 3x)^5$.

Find the derivative of $s(6x)$.

Find the derivative of $\cos(x^2)$.

Find the derivative of $\tan(x^3)$.

Find the derivative of $\sec(4x)$.

Find the derivative of $(x^3 - 7)^{12}$.

Using the chain rule, find $\frac{dz}{dt}$ for a function $z = f(x, y)$ where $x = x(t)$ and $y = y(t)$.

Given a function $z = x^3 + y^3$ where $x = 2\sin(t)$ and $y = 3\cos(t)$, calculate $\frac{dz}{dt}$.

What direction should you travel to increase your height on a mountain as fast as possible?

Compute the gradient of the function $f(x, y) = x^2 \sin(y)$.

Compute the gradient of a multivariable function by finding its partial derivatives and forming a vector.

Find the gradient of the function $f(x, y) = 3x^2 + y + 6$ at the point $(1, -1)$.

Find the gradient of a scalar function $z = f(x, y) = x^2 + y^2$, and evaluate it at the points (2, 1) and (-1, -1).

Find the gradient of the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ and evaluate it at the point $(-3, 2)$.

For the function $f(x, y) = x^2y - xy^2$ at the point (1, -1), find the direction and rate of greatest increase, greatest decrease, and a direction of no change.

Linearize the multivariable function $f(x,y) = 1 + x \ln(xy - 5)$ at the point (2, 3).