Question

Consider the expression w=xy ^2+x ^2z+yz ^2 , where x=t ^2, y=9t, and z=9. Find dt/dw by converting w to a function of t before differentiating.

Answer 1

To find dt/dw, we need to convert w into a function of t. By substituting the given expressions for x, y, and z into the expression for w and simplifying, we can obtain w as a function of t. Then we differentiate w with respect to t using the power rule to find dw/dt. Finally, we can substitute the given values for t to find dt/dw.

Step-by-step explanation:

To find dt/dw, we need to convert w into a function of t. Given the expressions for x, y, and z, we can substitute their values into the expression for w and simplify:

w = (t2)(9t)2 + (t2)2(9) + (9)(9t)2

Simplifying further, we have:

w = 81t4 + 9t4 + 729t2

Next, we can differentiate w with respect to t. Using the power rule, we get:

dw/dt = 324t3 + 36t3 + 1458t

Finally, we can substitute the given values for t (and any other variables) to find the value of dt/dw.