Differentiate implicitly to find the first partial derivatives of \( z \) \[ 3 x^{2}+7 y^{2}+5 z^{2}=25 \]

To find the first partial derivatives of \( z \) with respect to \( x \) and \( y \) from the given equation \( 3x^2 + 7y^2 + 5z^2 = 25 \), we can use implicit differentiation. For \( \frac{\partial z}{\partial x} \): 1. Differentiate both sides with respect to \( x \): \[ 6x + 0 + 10z \frac{\partial z}{\partial x} = 0 \] 2. Solving for \( \frac{\partial z}{\partial x} \) yields: \[ 10z \frac{\partial z}{\partial x} = -6x \Rightarrow \frac{\partial z}{\partial x} = -\frac{6x}{10z} = -\frac{3x}{5z} \] For \( \frac{\partial z}{\partial y} \): 1. Differentiate both sides with respect to \( y \): \[ 0 + 14y + 10z \frac{\partial z}{\partial y} = 0 \] 2. Solving for \( \frac{\partial z}{\partial y} \) gives: \[ 10z \frac{\partial z}{\partial y} = -14y \Rightarrow \frac{\partial z}{\partial y} = -\frac{14y}{10z} = -\frac{7y}{5z} \] Thus, the first partial derivatives are: \[ \frac{\partial z}{\partial x} = -\frac{3x}{5z} \quad \text{and} \quad \frac{\partial z}{\partial y} = -\frac{7y}{5z} \] Now, here's some additional information! Did you know that implicit differentiation is heavily used in varying fields like physics and engineering? When dealing with complicated equations that define relationships between variables, this technique helps in finding rates of change without having to isolate one variable. It's like putting on a pair of magical glasses that allow you to see the hidden relationships—so handy! If you’re curious about exploring more complex functions or systems where implicit differentiation might come in handy, consider delving into multivariable calculus resources. Books focusing on vector calculus often cover these concepts with applications that involve gradients, curl, and divergence, which can truly open up a world of understanding about the relationships between multiple changing variables!