Question

Use the chain rule to find dw/dt. w = xey/z, x = t9, y = 5 − t, z = 9 4t

Answer 1

`dw/dt = 72t^8 e^(t(5 - t)) / (81z^2)` differentiating with respect to t involves carefully applying the quotient rule and the chain rule for the exponential function. The final result is dw/dt =
`72t^8 * e^(t(5 - t)) / (81z^2),` encapsulating the rate of change of w concerning t with respect to the given functions

Explanation:

To find dw/dt using the chain rule, we first express w in terms of t by substituting the given expressions for x, y, and z into w = xey/z. Given x = t^9, y = 5 - t, and z = 9/(4t), we substitute these values into w, obtaining w =
`t^9 * e^(t(5 - t)) / (9/(4t))`. Simplifying, we get w =
`4t^10 * e^(5t - t^2) / 9.`To differentiate w with respect to t, we apply the chain rule, yielding dw/dt = d/dt [4t^10 * e^(5t - t^2) / 9]. Using the quotient rule and the chain rule for the exponential function, the derivative simplifies to dw/dt =
`72t^8 * e^(t(5 - t)) / (81z^2), where z = 9/(4t).`

This differentiation involves applying the chain rule to handle the composition of functions within the expression for w. Initially, we substitute the given expressions into w, forming an equation in terms of t. From there, differentiating with respect to t involves carefully applying the quotient rule and the chain rule for the exponential function. The final result is dw/dt =
`72t^8 * e^(t(5 - t)) / (81z^2),` encapsulating the rate of change of w concerning t with respect to the given functions. This method allows us to effectively compute derivatives involving multiple variables and functions.