Question

Find \(d w / d t\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(t\) before differentiating. \(w=x^{2}+y^{2}+z^{2}, \quad x=e^{t} \cos t, \quad y=e^{t} \sin t, \quad z=e^{t}\)

Short answer

Therefore, by both methods, we get \(dw / dt\) as \(3e^{2t}(1 + cos(2t))\).

Step-by-step solution

Apply the Chain Rule

To begin, we use the chain rule to find \(dw / dt\). Since \(w = x^{2} + y^{2} + z^{2}\), where \(x\), \(y\), and \(z\) are each functions of \(t\), we express \(dw / dt\) as \(\frac{dw}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt} + 2z\frac{dz}{dt}\)

Substitute the Expressions for \(x\), \(y\) and \(z\)

Next, we substitute \(x = e^{t}cos(t)\), \(y = e^{t}sin(t)\), and \(z = e^{t}\) into the formula from step 1. We also find each of \(\frac{dx}{dt}\), \(\frac{dy}{dt}\), and \(\frac{dz}{dt}\) using basic rules of differentiation.

Simplify the Expression for \(dw/dt\)

After inserting the computed values of \(\frac{dx}{dt}\), \(\frac{dy}{dt}\), and \(\frac{dz}{dt}\), as well as \(x\), \(y\) and \(z\) into the formula, we simplify the expression to get the final answer for part a.

Convert \(w\) into a Function of \(t\)

For part b, we need to convert \(w\) into a function of \(t\) before differentiating. We substitute the given functions \(x = e^{t}cos(t)\), \(y = e^{t}sin(t)\), and \(z = e^{t}\) into \(w = x^{2} + y^{2} + z^{2}\).

Differentiate \(w\) with Respect to \(t\)

After simplifying \(w\) into a function of \(t\), we differentiate the resulting function with respect to \(t\) to get an expression for \(\frac{dw}{dt}\).