Partial Differentiation with Exercises and Solutions
Definition of Derivative
Example 1: Derivative of f(x) with respect to x:
\[
\frac{df}{dx}(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
\]
Partial Differentiation
Partial derivative of f(x,y) with respect to x:
\[
\frac{\partial f}{\partial x}(x, y)
= \lim_{\Delta x \to 0}
\frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}
\]
Partial derivative of f(x,y) with respect to y:
\[
\frac{\partial f}{\partial y}(x, y)
= \lim_{\Delta y \to 0}
\frac{f(x, y + \Delta y) - f(x, y)}{\Delta y}
\]
Example 1:
\[
f(x, y) = e^{-(x^{2} + y^{2})}
\]
Question: Find the partial derivative with respect to x.
Solution 1
View the solutionExample 2:
\[
f(x, y) = -(x-3)^{2} + xy + 16
\]
Find the partial derivative with respect to x.
Solution 2
View the solutionPartial differentiation: Practical Questions
Example 3: Calculate all first and second order partial derivatives of:
\[
f(x, y) = e^{(3x^{2}y)}
\]
Solution 3
View the solutionAdditional partial derivative with respect to x and y
Example 4:
\[
f(x, y) = x / y
\]
Solution 4
View the solutionExample 5:
\[
f(x, y) = - (x-3y)^{2}
\]
Solution 5
View the solutionExample 6:
\[
f(x, y) = e^{(2x+y^{3})}
\]
Solution 6
View the solutionExample 7:
\[
f(x, y) = xln(2y)
\]
Solution 7
View the solutionExample 8:
\[
f(x, y) = cos(x(e^{3x}))
\]