Calculus 3

Find the arc length of the vector-valued function $\mathbf{R}(t) = 3t\mathbf{i} - t\mathbf{j}$ over the interval \([0, 3]\).

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 $(\ln x)^7$.

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

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

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

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

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

Calculate the dot product of vectors $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

Calculate the dot product of vectors $\mathbf{a} = (3, -4, 7)$ and $\mathbf{b} = (5, 2, -3)$.

Calculate the square of the magnitude of vector $\mathbf{a} = (2, 3)$.

Calculate the dot product of $a$ and $b$ times vector $a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

Calculate the dot product between vector $b$ and $3a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.