Introduction to Partial Derivatives
Partial derivatives are used to measure how a function changes as one of its variables changes, while keeping other variables constant. They are essential in multivariable calculus and are widely used in fields such as physics, engineering, and economics.
Mathematical Formulation
For a function \( f(x, y) \), the partial derivative with respect to \( x \) is:
$$ \frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x} $$
This represents the rate of change of the function \( f \) with respect to \( x \), while keeping \( y \) constant.
Simple Example
Consider \( f(x, y) = x^2y + y^3 \). The partial derivative with respect to \( x \) is:
$$ \frac{\partial f}{\partial x} = 2xy $$
And with respect to \( y \):
$$ \frac{\partial f}{\partial y} = x^2 + 3y^2 $$
Intermediate Example
Let's find the partial derivatives of \( g(x, y, z) = x^2y + yz + z^3 \).
With respect to \( x \):
$$ \frac{\partial g}{\partial x} = 2xy $$
With respect to \( y \):
$$ \frac{\partial g}{\partial y} = x^2 + z $$
With respect to \( z \):
$$ \frac{\partial g}{\partial z} = y + 3z^2 $$
Advanced Example
Consider a more complex function \( h(x, y) = e^{xy} \sin(x + y) \).
To find \( \frac{\partial h}{\partial x} \), use the product rule and chain rule:
$$ \frac{\partial h}{\partial x} = e^{xy} \cdot y \cdot \sin(x + y) + e^{xy} \cdot \cos(x + y) $$
To find \( \frac{\partial h}{\partial y} \), similarly apply the product rule and chain rule:
$$ \frac{\partial h}{\partial y} = e^{xy} \cdot x \cdot \sin(x + y) + e^{xy} \cdot \cos(x + y) $$
Visualizing Partial Derivatives
Imagine a surface in three-dimensional space. The partial derivative with respect to \( x \) at a point on the surface represents the slope of the tangent line in the \( x \)-direction, while keeping \( y \) constant. Similarly, the partial derivative with respect to \( y \) represents the slope in the \( y \)-direction.
Applications in Real Life
Partial derivatives are used in various fields to model and analyze systems with multiple variables. In physics, they are used to describe how physical quantities change with respect to each other. In economics, they help in understanding how changes in one economic variable affect others.