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The gradient of a function is an essential concept in calculus that acts like a compass pointing in the direction of the greatest rate of increase of the function. For multivariable functions, the gradient is a vector that consists of all its first partial derivatives. When you have a function like the one given, \[ f(x, y, z) = x y z - 10 \] we calculate each of the partial derivatives. These are:

• \( f_x = yz \) - this is the derivative of \( f \) with respect to \( x \)
• \( f_y = xz \) - this is the derivative of \( f \) with respect to \( y \)
• \( f_z = xy \) - this is the derivative of \( f \) with respect to \( z \)

When you find these partial derivatives, you essentially peel apart how each variable affects the function's behavior around a given point. To find the gradient, you gather these partial derivatives into a vector: \[ abla f = \langle f_x, f_y, f_z \rangle \]. At the given point (1, 2, 5), substitute the values into the partial derivatives to get the gradient vector: \[ abla f(1, 2, 5) = \langle 10, 5, 2 \rangle \]. This vector is perpendicular, or normal, to the surface at the given point.

The normal line to a surface at a given point is a line that is perpendicular to the tangent plane of the surface at that point. When finding the normal line, the key is to use the gradient vector, \( abla f \), calculated from the previous section. This gradient directly acts as the direction vector for our normal line. To express the symmetric equations of the normal line, you take the formula: \[ \frac{x - x_0}{A} = \frac{y - y_0}{B} = \frac{z - z_0}{C} \]. Here, \( (x_0, y_0, z_0) \) is the point, \( (1, 2, 5) \), and \( A, B, C \) are the components of the gradient vector \( \langle 10, 5, 2 \rangle \). So, after substituting these into the equation, we get: \[ \frac{x - 1}{10} = \frac{y - 2}{5} = \frac{z - 5}{2} \]. This describes a line in three-dimensional space that is normal to the surface at the point (1, 2, 5). It extends infinitely in both directions, providing a clear path along the steepest ascent from the surface.

Partial derivatives are used to measure how a function changes as only one of the input variables changes, while others are kept constant. For a function of multiple variables such as \( f(x, y, z) \), you calculate the partial derivatives with respect to each variable: \( x \), \( y \), and \( z \). This is similar to finding a derivative in single-variable calculus but applied to each variable individually.

• For \( x \), the partial derivative is \( f_x = yz \), meaning it shows how changes in \( x \) affect \( f \), with \( y \) and \( z \) constant.
• For \( y \), the partial derivative is \( f_y = xz \), indicating how changes in \( y \) affect \( f \) while holding \( x \) and \( z \) steady.
• For \( z \), it is \( f_z = xy \), revealing how changes in \( z \) impact \( f \) when \( x \) and \( y \) remain unchanged.

Partial derivatives help formulate the gradient vector, \( abla f \), discussed earlier. They illustrate the sensitivity of the function to each variable and pave the way for understanding changes along curves, surfaces, and volumes in multidimensional spaces. Calculating these partial derivatives is critical for analyzing surfaces and establishing tangent planes.