To find the differential of the function \(f(t, u, v, w) = t v^6 u v w\), we need to understand how to apply the rules of differentiation with respect to multiple variables.
The differential of a function \(f\) with respect to its variables can be expressed as:
\[ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial u} du + \frac{\partial f}{\partial v} dv + \frac{\partial f}{\partial w} dw \]
Now, let’s calculate each of the partial derivatives:
1. Calculate \(\frac{\partial f}{\partial t}\)
Since \(f(t, u, v, w) = t v^6 u v w\) is linear in \(t\), the partial derivative is:
\[ \frac{\partial f}{\partial t} = v^6 u v w \]
2. Calculate \(\frac{\partial f}{\partial u}\)
Next, we treat all other variables as constants when differentiating with respect to \(u\):
\[ \frac{\partial f}{\partial u} = t v^6 v w \]
3. Calculate \(\frac{\partial f}{\partial v}\)
Here, we need to use the product rule since \(v\) appears multiple times in the function.
Applying the product rule, we get:
\[ \frac{\partial f}{\partial v} = t u w \left(6v^5 \cdot v + v^6 u\right) = t u w (6v^6 + v^6 u) = t u w v^6 (6 + u) \]
4. Calculate \(\frac{\partial f}{\partial w}\)
Finally, treating all other variables as constants, the partial derivative is:
\[ \frac{\partial f}{\partial w} = t v^6 u v \]
5. Substitute the partial derivatives into the differential formula
Now that we have calculated all of the partial derivatives, we can substitute them into the differential formula:
\[ df = (v^6 u v w) dt + (t v^6 v w) du + (t u w v^6 (6 + u)) dv + (t v^6 u v) dw \]
Therefore, the final expression for the differential of the function \(f(t, u, v, w) = t v^6 u v w\) is:
\[ df = v^6 u v w \, dt + t v^6 v w \, du + t u w v^6 (6 + u) \, dv + t v^6 u v \, dw \]
This gives you a complete view of how the function changes in response to infinitesimal changes in each of its variables.