Question

step 1 using the diagram of a right triangle given below, the relation between x, y, and z is $z^{2}=x^{2}+y^{2}$ step 2 we must find dz/dt. differentiating both sides and simplifying gives us the following. $\frac{dz}{dt}=2x\cdot\frac{dx}{dt}+\cdot\frac{dy}{dt}$ $\frac{dz}{dt}=(x\cdot\frac{dx}{dt}+\cdot\frac{dy}{dt})$

Explanation:

Step1: Apply chain - rule to differentiate $z^{2}$

Differentiating $z^{2}$ with respect to $t$ gives $2z\frac{dz}{dt}$.

Step2: Differentiate $x^{2}+y^{2}$ with respect to $t$

Using the sum - rule and chain - rule, 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, $2z\frac{dz}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$. Then, dividing both sides by $2z$ gives $\frac{dz}{dt}=\frac{1}{z}(x\frac{dx}{dt}+y\frac{dy}{dt})$.

Answer:

The first blank in the first equation of Step 2 is $2z$, the second blank is $2y$, the first blank in the second equation of Step 2 is $\frac{1}{z}$, and the second blank is $y$.