What is a Partial Derivative?
For a function f(x, y), the partial derivative with respect to x represents the rate at which f changes as x changes, keeping y constant. Similarly, the partial derivative with respect to y measures the rate at which f changes as y changes, keeping x constant.
Notation and Definition
Partial Derivative with respect to x:
\frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y, z) - f(x, y, z)}{\Delta x}
Partial Derivative with respect to y:
\frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y, z) - f(x, y, z)}{\Delta y}
Partial Derivative with respect to z:
\frac{\partial f}{\partial z} = \lim_{\Delta z \to 0} \frac{f(x, y, z + \Delta z) - f(x, y, z)}{\Delta z}
The partial derivative of f with respect to x is denoted by f_x or \frac{\partial f}{\partial x}.
The partial derivative of f with respect to y is denoted by f_y or \frac{\partial f}{\partial y}.
Properties of Partial Derivatives:
1. Linearity:
If f and g are functions of x and y, and a and b are constants, then:
\frac{\partial}{\partial x} (af + bg) = a \frac{\partial f}{\partial x} + b \frac{\partial g}{\partial x}
2. Product Rule
If u and v are functions of x and y, then:
\frac{\partial}{\partial x} (uv) = u \frac{\partial v}{\partial x} + v \frac{\partial u}{\partial x}
3. Quotient Rule
If u and v are functions of x and y, and v \neq 0, then:
\frac{\partial}{\partial x} \left( \frac{u}{v} \right) = \frac{v \frac{\partial u}{\partial x} - u \frac{\partial v}{\partial x}}{v^2}
4. Chain Rule
If z = f(u, v), where u and v are functions of x and y, then:
\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}
5. Higher Order Partial Derivatives
The second-order partial derivative with respect to x is denoted as:
\frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right)
Mixed partial derivatives involve derivatives with respect to different variables:
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right)
6. Clairaut’s Theorem (Symmetry of Mixed Partial Derivatives)
If f is continuously differentiable, then the mixed partial derivatives are equal:
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}
Applications of partial derivative:
1. Gradient:
The gradient of a scalar function f(x, y, z) is a vector of its first partial derivatives:
\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)
2. Directional Derivatives:
The directional derivative of f in the direction of a unit vector \mathbf{u} = (u_x, u_y, u_z) is given by:
D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} = \frac{\partial f}{\partial x} u_x + \frac{\partial f}{\partial y} u_y + \frac{\partial f}{\partial z} u_z
3. Tangent Plane and Normal Line:
For a function z = f(x, y), the equation of the tangent plane at point (x_0, y_0, z_0) is:
z - z_0 = \frac{\partial f}{\partial x} (x_0, y_0) (x - x_0) + \frac{\partial f}{\partial y} (x_0, y_0) (y - y_0)
The normal line can be derived from the gradient at that point
Example: Calculating Partial Derivatives
Consider the function f(x,y)=x^2y+y^3. Let’s find the first partial derivatives \frac{\partial f}{\partial x} and \frac{\partial f}{\partial y}.
f(x,y)=x^2y+y^3
Take the partial derivative with respect to x.
\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}\left(x^2y+y^3\right)\\
\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}\left(x^2y\right)+\frac{\partial }{\partial x}\left(y^3\right)\\
\frac{\partial f}{\partial x}=y\frac{\partial }{\partial x}\left(x^2\right)+0\\
\frac{\partial f}{\partial x}=y(2x)+0\\
\frac{\partial f}{\partial x}=2xy\\
.
f(x,y)=x^2y+y^3
Take the partial derivative with respect to y.
\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}\left(x^2y+y^3\right)\\
\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}\left(x^2y\right)+\frac{\partial }{\partial y}\left(y^3\right)\\
\frac{\partial f}{\partial y}=x^2\frac{\partial }{\partial y}\left(y\right)+3y^2\\
\frac{\partial f}{\partial y}=x^2\cdot 1+3y^2\\
\frac{\partial f}{\partial y}=x^2+3y^2\\
Example: Find the equation of the tangent plane to the surface z=f(x,y)=10-x^2-2y^2 at the point (1,2).
Step-1: Evaluate f at the point (x_0, y_0)=(1,2):
z_0=10-1^2-2(2)^2\\
z_0=10-1-8\\
z_0=1\\
The point on the surface is (x_0, y_0, z_0)=(1,2,1).
Step-2: Find the partial derivatives f_x.
f_x=\frac{\partial }{\partial x}\left(10-x^2-2y^2\right)\\
f_x=\frac{\partial }{\partial x}\left(10\right)-\frac{\partial }{\partial x}\left(x^2\right)-\frac{\partial }{\partial x}\left(2y^2\right)\\
f_x=0-2x-0\\
f_x=-2x\\
Substitute the given point (1,2).
f_x=-2(1)\\
f_x=-2\\
Step-3: Find the partial derivatives f_y.
f_y=\frac{\partial }{\partial y}\left(10-x^2-2y^2\right)\\
f_y=\frac{\partial }{\partial y}\left(10-x^2-2y^2\right)\\
f_y=\frac{\partial }{\partial y}\left(10\right)-\frac{\partial }{\partial y}\left(x^2\right)-\frac{\partial }{\partial y}\left(2y^2\right)\\
f_y=0-0-4y\\
f_y=-4y\\
Substitute the given point (1,2).
f_y=-4(2)\\
f_y=-8\\
Step-4: Write the equation of the tangent plane.
The equation of the tangent plane is given by,
z=z_0+f_x(x-x_0)+f_y(y-y_0)\\
z=1-2(x-1)-8(y-2)\\
z=1-2x+2-8y+16\\
z=-2x-8y+19\\
2x+8y+z=19\\
The equation of the tangent plane is 2x+8y+z=19\\.
FAQs on Partial Derivatives:
1) What is a partial derivative?
Answer: A partial derivative is a mathematical concept that measures how a function changes with respect to one of its variables while keeping the other variables constant. It is denoted by \frac{\partial f}{\partial x} or f_x and represents the rate of change of the function in the direction of the x-axis.
2) When are partial derivatives used?
Answer: Partial derivatives are used in multivariable calculus to analyze the behavior of functions of several variables. They are essential in optimization problems, where finding maxima and minima requires understanding how a function changes with respect to each variable independently.
3) How do you calculate a partial derivative?
To calculate a partial derivative, you treat all variables except the one you are differentiating with respect to as constants and take the derivative as you would with a single-variable function. For example, to find \frac{\partial f}{\partial x}, you differentiate the function f(x,y) with respect to x while treating y as a constant.
4) What is ∂ called?
Answer: The symbol ∂ is called “partial derivative” or “partial differential operator.”