Find the first partial derivatives of the function: $u = 5x^{\frac{y}{z}}$

Neetesh Kumar | December 3, 2024

Calculus Homework Help

This is the solution to Math 1D
Assignment: 14.3 Question Number 19
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Step-by-step solution:

We are given the function: $u = 5x^{\frac{y}{z}}$

To find the partial derivatives, we will differentiate with respect to each variable while treating the other variables as constants.

Step 1: Find $\frac{\partial u}{\partial x}$

We will differentiate $u = 5x^{\frac{y}{z}}$ with respect to $x$. Use the power rule and treat $y$ and $z$ as constants:

$\frac{\partial u}{\partial x} = 5 \cdot \frac{y}{z} x^{\frac{y}{z} - 1}.$

Thus:

$\frac{\partial u}{\partial x} = \frac{5y}{z} x^{\frac{y}{z} - 1}.$

Step 2: Find $\frac{\partial u}{\partial y}$

Next, we differentiate with respect to $y$, treating $x$ and $z$ as constants. We will apply the chain rule:

$\frac{\partial u}{\partial y} = 5 \cdot \frac{\partial}{\partial y} \left( x^{\frac{y}{z}} \right).$

The derivative of $x^{\frac{y}{z}}$ with respect to $y$ is:

$\frac{\partial}{\partial y} \left( x^{\frac{y}{z}} \right) = \frac{x^{\frac{y}{z}} \ln(x)}{z}.$

Thus:

$\frac{\partial u}{\partial y} = 5 \cdot \frac{x^{\frac{y}{z}} \ln(x)}{z}.$

Step 3: Find $\frac{\partial u}{\partial z}$

Now, we differentiate with respect to $z$, treating $x$ and $y$ as constants. Again, we use the chain rule:

$\frac{\partial u}{\partial z} = 5 \cdot \frac{\partial}{\partial z} \left( x^{\frac{y}{z}} \right).$

The derivative of $x^{\frac{y}{z}}$ with respect to $z$ is:

$\frac{\partial}{\partial z} \left( x^{\frac{y}{z}} \right) = -\frac{y x^{\frac{y}{z}} \ln(x)}{z^2}.$

Thus:

$\frac{\partial u}{\partial z} = -\frac{5y x^{\frac{y}{z}} \ln(x)}{z^2}.$

Final Answer:

$\frac{\partial u}{\partial x} = \boxed{\frac{5y}{z} x^{\frac{y}{z} - 1}}$

$\frac{\partial u}{\partial y} = \boxed{\frac{5 x^{\frac{y}{z}} \ln(x)}{z}}$

$\frac{\partial u}{\partial z} = \boxed{-\frac{5y x^{\frac{y}{z}} \ln(x)}{z^2}}$

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