Partial Differentiation : An Introduction.

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Introduction to Partial Differentiation

In many practical situations, a quantity depends on more than one independent variable. For example, the volume of a gas may depend on both pressure and temperature, or the area of a surface may depend on two spatial coordinates. Such functions are called functions of several variables.

If a function depends on two independent variables $x$ and $y$, it is written as $$ z = f(x, y) $$

Partial differentiation is the process of finding the rate of change of a function with respect to one variable while keeping the other variable(s) constant.

The partial derivative of $z$ with respect to $x$ is denoted by $$ \frac{\partial z}{\partial x} $$ and is obtained by differentiating $z$ with respect to $x$, treating $y$ as a constant.

Similarly, the partial derivative of $z$ with respect to $y$ is denoted by $$ \frac{\partial z}{\partial y} $$ and is obtained by differentiating $z$ with respect to $y$, treating $x$ as a constant.

For example, if $$ z = x^2y + xy^2 $$ then $$ \frac{\partial z}{\partial x} = 2xy + y^2, \quad \frac{\partial z}{\partial y} = x^2 + 2xy $$

Partial differentiation plays an important role in mathematics, physics, engineering, economics, and other applied sciences. It is widely used in topics such as total derivatives, maxima and minima, Euler’s theorem, and differential equations.

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Partial Differentiation Problems with Solutions

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Problem 1

If $z = x^2y + xy^3$, find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy + y^3 $$ $$ \frac{\partial z}{\partial y} = x^2 + 3xy^2 $$

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Problem 2

If $z = x^3 + 3xy^2$, find $\frac{\partial^2 z}{\partial x^2}$ and $\frac{\partial^2 z}{\partial y^2}$.

Solution:

$$ \frac{\partial z}{\partial x} = 3x^2 + 3y^2 \Rightarrow \frac{\partial^2 z}{\partial x^2} = 6x $$ $$ \frac{\partial z}{\partial y} = 6xy \Rightarrow \frac{\partial^2 z}{\partial y^2} = 6x $$

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Problem 3

For $z = x^2y^3$, verify that $$ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} $$

Solution:

$$ \frac{\partial z}{\partial x} = 2xy^3 \Rightarrow \frac{\partial^2 z}{\partial y \partial x} = 6xy^2 $$ $$ \frac{\partial z}{\partial y} = 3x^2y^2 \Rightarrow \frac{\partial^2 z}{\partial x \partial y} = 6xy^2 $$

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Problem 4

If $x^2 + y^2 + z^2 = a^2$, find $\frac{\partial z}{\partial x}$.

Solution:

$$ 2x + 2z\frac{\partial z}{\partial x} = 0 $$ $$ \frac{\partial z}{\partial x} = -\frac{x}{z} $$

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Problem 5

If $z = x^2 + y^2$, where $x = r\cos\theta$ and $y = r\sin\theta$, find $\frac{\partial z}{\partial r}$.

Solution:

$$ z = r^2(\cos^2\theta + \sin^2\theta) = r^2 $$ $$ \frac{\partial z}{\partial r} = 2r $$

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Problem 6 (Euler’s Theorem)

Verify Euler’s theorem for $z = x^2y + xy^2$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy + y^2 $$ $$ \frac{\partial z}{\partial y} = x^2 + 2xy $$ $$ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 3x^2y + 3xy^2 = 3z $$

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Problem 7

If $z = \ln(x^2 + y^2)$, find $\frac{\partial z}{\partial x}$.

Solution:

$$ \frac{\partial z}{\partial x} = \frac{2x}{x^2 + y^2} $$

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Problem 8

If $z = e^{xy}\sin y$, find $\frac{\partial z}{\partial y}$.

Solution:

$$ \frac{\partial z}{\partial y} = e^{xy}(x\sin y + \cos y) $$

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Problem 9

If $z = x^2y + y^2$, find the total differential $dz$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy,\quad \frac{\partial z}{\partial y} = x^2 + 2y $$ $$ dz = 2xy\,dx + (x^2 + 2y)\,dy $$

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Problem 10

Find $\frac{\partial}{\partial x}(x^2y^3)$.

Solution:

$$ \frac{\partial}{\partial x}(x^2y^3) = 2xy^3 $$

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📘 End of Exam-Oriented Problems