question 4 if z² = x² + y², then dz/dt = 2x + 2y dy/dt. true false question 5

Explanation:

Step1: Differentiate both sides with respect to $t$

Using the chain - rule, if $z^{2}=x^{2}+y^{2}$, then $\frac{d}{dt}(z^{2})=\frac{d}{dt}(x^{2}+y^{2})$. The derivative of $z^{2}$ with respect to $t$ is $2z\frac{dz}{dt}$ by the chain - rule ($\frac{d}{du}(u^{2})\cdot\frac{du}{dt}$ where $u = z$), the derivative of $x^{2}$ with respect to $t$ is $2x\frac{dx}{dt}$ and the derivative of $y^{2}$ with respect to $t$ is $2y\frac{dy}{dt}$. So, $2z\frac{dz}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$, and $\frac{dz}{dt}=\frac{x\frac{dx}{dt}+y\frac{dy}{dt}}{z}$.

Answer:

B. false