Engineering Mathematics: Finding the Partial Derivative of a Given Function with Respect to x

To find the partial derivative of a given function with respect to x, we can use the product rule of differentiation. The function given is:

f(x,y,z) = (x²/a² u) (y²/b² u) (z²/c² u)

To find du/dx, we need to differentiate f with respect to x while treating u as a constant.

Using the Product Rule of Differentiation

We can use the product rule of differentiation to find the partial derivative of f with respect to x. The product rule states:

(d/dx) [f(x)g(x)] = f'(x) g(x) + f(x) g'(x)

Where f(x) and g(x) are functions of x, and f'(x) and g'(x) are their derivatives with respect to x.

Breaking Down the Function

We can break down the given function as follows:

f(x,y,z) = (x²/a² u) (y²/b² u) (z²/c² u)
= x²y²z²/(a²b²c²) u³

Partial Derivative with Respect to x

Applying the product rule of differentiation, we get:

du/dx = (d/dx) [(x²y²z²)/(a²b²c²) u³]
= (2xy²z²)/(a²b²c²) u³

Therefore, the partial derivative of f with respect to x is:

du/dx = (2xy²z²)/(a²b²c²) u³

Conclusion

To find the partial derivative of a function with respect to a variable, we need to apply the product rule of differentiation. In this case, we used the product rule to find the partial derivative of the given function with respect to x. The final result is (2xy²z²)/(a²b²c²) u³.