Question

Use The Chain Rule To Find Dw/Dt. W=Ln(X2+Y2+Z2), X=6sin(T), Y=5cos(T), Z=7tan(T)

Dt/dw=________

Answer 1

The process involves using the chain rule to differentiate
`w = ln(x^2+y^2+z^2)` with the given functions for x, y, and z, which are functions of t. By calculating the partial derivatives of w and multiplying by the derivatives of x, y, and z with respect to t, we obtain dw/dt.

Step-by-step explanation:

To find dw/dt using the chain rule, we need to differentiate the function w with respect to t. We have w = ln(x2+y2+z2), where x, y, and z are functions of t, specifically x = 6sin(t), y = 5cos(t), and z = 7tan(t).

First, we differentiate w with respect to x, y, and z, and then multiply each by the derivative of x, y, and z with respect to t:

dw/dx = 1/(x2+y2+z2) × 2x

dw/dy = 1/(x2+y2+z2) × 2y

dw/dz = 1/(x2+y2+z2) × 2z

Now, we find the derivatives of x, y, and z with respect to t:

dx/dt = 6cos(t)

dy/dt = -5sin(t)

dz/dt = 7sec2(t)

Finally, we apply the chain rule to combine these derivatives:

dw/dt = (dw/dx)(dx/dt) + (dw/dy)(dy/dt) + (dw/dz)(dz/dt)

dw/dt = (2x/(x2+y2+z2))× 6cos(t) + (2y/(x2+y2+z2))× -5sin(t) + (2z/(x2+y2+z2))× 7sec2(t)

By substituting the given functions x, y, and z into this equation, you would obtain the value for dw/dt.