z=u(v+w)^(1/2) find ∂^3z/(∂u∂v∂w)
what partial derivative am I suppose to be finding exactly? Do I take the derivative of z three times?
Please help. Thank you.
To find ∂^3z/(∂u∂v∂w), you need to take the derivative of z three times, once with respect to u, once with respect to v, and once with respect to w.
Here's how you can calculate it step by step:
1. **First partial derivative**: Start by taking the partial derivative of z with respect to u. Treat v and w as constants when differentiating with respect to u. The square root term can be rewritten as (v+w)^(1/2) = (v+w)^(0.5). Differentiating z with respect to u gives:
∂z/∂u = ∂(u(v+w)^(0.5))/∂u = (v+w)^(0.5).
2. **Second partial derivative**: Taking the partial derivative of the result from step 1 with respect to v will give the second partial derivative. Differentiating (v+w)^(0.5) with respect to v gives:
∂^2z/(∂u∂v) = ∂((v+w)^(0.5))/∂v = 0.5(u(v+w)^(-0.5))(1) = 0.5u(v+w)^(-0.5).
3. **Third partial derivative**: Finally, take the partial derivative of the result from step 2 with respect to w to obtain the third partial derivative. Differentiating 0.5u(v+w)^(-0.5) with respect to w gives:
∂^3z/(∂u∂v∂w) = ∂(0.5u(v+w)^(-0.5))/∂w = -0.25u(v+w)^(-1.5).
So, the third partial derivative of z with respect to u, v, and w is -0.25u(v+w)^(-1.5).