Total Derivatives part 1

Geometric Interpretation of Total Derivative

Consider a function of two variables $$ z = f(x, y) $$ This function represents a surface in three-dimensional space. Each point on the surface corresponds to a particular value of $x$ and $y$.

x z y (x, y, z)

The red point represents a point on the surface corresponding to the values $(x, y)$.

Small Changes in Variables

Suppose the independent variables $x$ and $y$ change by small amounts $dx$ and $dy$. These small changes cause a change in $z$, denoted by $dz$.

dx dy

The movement along the $x$-direction by $dx$ and along the $y$-direction by $dy$ together produce a total change in the value of $z$.

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Total Change in z

The total change in $z$ is the combined effect of the change due to $x$ and the change due to $y$. Mathematically, it is given by:

$$ dz = \frac{\partial z}{\partial x}\,dx + \frac{\partial z}{\partial y}\,dy $$

dz

Thus, the total derivative represents the net rate of change of the function due to simultaneous changes in all independent variables.

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Introduction to Total Derivatives

When a dependent variable depends on more than one independent variable, its change depends on the changes in all those variables. In such cases, we use the concept of Total Derivative.

If a function is given by $$ z = f(x, y) $$ where $x$ and $y$ are independent variables, then a small change in $z$ due to small changes in $x$ and $y$ is called the total differential.

The total differential of $z$ is denoted by $dz$ and is defined as: $$ dz = \frac{\partial z}{\partial x}\,dx + \frac{\partial z}{\partial y}\,dy $$

Total derivatives are widely used in error analysis, approximation, thermodynamics, and engineering applications.

Examples on Total Derivatives

Example 1

Find the total differential of $z = x^2y$.

Solution:

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

Example 2

Find $dz$ if $z = x^2 + y^2$.

Solution:

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

Example 3

Find the total differential of $z = xy + y^2$.

Solution:

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

Example 4

Find $dz$ if $z = x^3y^2$.

Solution:

$$ \frac{\partial z}{\partial x} = 3x^2y^2, \quad \frac{\partial z}{\partial y} = 2x^3y $$ $$ dz = 3x^2y^2\,dx + 2x^3y\,dy $$

Example 5

Find the total derivative of $z = \ln(x^2 + y^2)$.

Solution:

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

Example 6

Find $dz$ if $z = e^{xy}$.

Solution:

$$ \frac{\partial z}{\partial x} = ye^{xy}, \quad \frac{\partial z}{\partial y} = xe^{xy} $$ $$ dz = ye^{xy}\,dx + xe^{xy}\,dy $$

Example 7

Find the total differential of $z = \sin(xy)$.

Solution:

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

Example 8

Find $dz$ if $z = x^2y + y^3$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy, \quad \frac{\partial z}{\partial y} = x^2 + 3y^2 $$ $$ dz = 2xy\,dx + (x^2 + 3y^2)\,dy $$

Example 9

Find the total differential of $z = \sqrt{x^2 + y^2}$.

Solution:

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

Example 10

Find $dz$ if $z = x^2y^2$.

Solution:

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

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Question 1

If $z = x^2y + y^2x$, find the total differential $dz$ and hence find the approximate change in $z$ when $x$ changes from 1 to 1.02 and $y$ changes from 2 to 2.01.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy + y^2, \quad \frac{\partial z}{\partial y} = x^2 + 2xy $$ At $x=1$, $y=2$: $$ \frac{\partial z}{\partial x} = 8,\quad \frac{\partial z}{\partial y} = 5 $$

$$ dx = 0.02,\quad dy = 0.01 $$

$$ dz = 8(0.02) + 5(0.01) = 0.21 $$

Question 2

If $z = \ln(x^2 + y^2)$, find the total differential and evaluate $dz$ at $x=2$, $y=1$.

Solution:

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

At $x=2$, $y=1$: $$ dz = \frac{4}{5}dx + \frac{2}{5}dy $$

Question 3

Find the total derivative of $z = e^{xy}$ and hence find the approximate change in $z$ when $x=1$, $y=2$, $dx=0.01$, $dy=0.02$.

Solution:

$$ \frac{\partial z}{\partial x} = ye^{xy}, \quad \frac{\partial z}{\partial y} = xe^{xy} $$

At $x=1$, $y=2$: $$ dz = 2e^2(0.01) + e^2(0.02) = 0.04e^2 $$

Question 4

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

Solution:

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

$$ dz = \frac{x}{\sqrt{x^2 + y^2}}dx + \frac{y}{\sqrt{x^2 + y^2}}dy $$

Question 5

If $z = x^3y^2$, find the total differential and hence find the rate of change of $z$ with respect to time $t$, given $x = t^2$ and $y = t$.

Solution:

$$ \frac{\partial z}{\partial x} = 3x^2y^2, \quad \frac{\partial z}{\partial y} = 2x^3y $$

$$ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} $$

Since $x=t^2$, $y=t$: $$ \frac{dx}{dt}=2t,\quad \frac{dy}{dt}=1 $$

$$ \frac{dz}{dt} = 3(t^4)(t^2)(2t) + 2(t^6)(t) $$

Question 6

Find the total differential of $z = \sin(xy)$.

Solution:

$$ \frac{\partial z}{\partial x} = y\cos(xy), \quad \frac{\partial z}{\partial y} = x\cos(xy) $$

$$ dz = y\cos(xy)\,dx + x\cos(xy)\,dy $$

Question 7

If $z = x^2y + y^3$, find $dz$ and hence find the approximate change in $z$ when $x=2$, $y=1$, $dx=0.05$, $dy=0.02$.

Solution:

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

At $x=2$, $y=1$: $$ dz = 4(0.05) + 7(0.02) = 0.34 $$

Question 8

Show that the total differential of $z = \ln(xy)$ is $$ dz = \frac{dx}{x} + \frac{dy}{y} $$

Solution:

$$ \frac{\partial z}{\partial x} = \frac{1}{x}, \quad \frac{\partial z}{\partial y} = \frac{1}{y} $$

$$ dz = \frac{dx}{x} + \frac{dy}{y} $$

Question 9

If $z = x^2 + y^2$, find the total differential and hence obtain the approximate error in $z$ when the errors in $x$ and $y$ are $\delta x$ and $\delta y$ respectively.

Solution:

$$ dz = 2x\,dx + 2y\,dy $$

Approximate error: $$ \delta z = 2x\,\delta x + 2y\,\delta y $$

Question 10

Find the total derivative of $z = x^2y^2$ and hence find the rate of change of $z$ with respect to $t$, given $x = \sin t$ and $y = \cos t$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy^2, \quad \frac{\partial z}{\partial y} = 2x^2y $$

$$ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} $$

📘 End of Ten-Marks Questions on Total Derivatives

📘 End of Total Derivative Examples

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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

Problem 1

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

Solution:

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

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:

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

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:

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

Problem 4

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

Solution:

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

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:

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

Problem 6 (Euler’s Theorem)

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

Solution:

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

Problem 7

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

Solution:

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

Problem 8

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

Solution:

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

Problem 9

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

Solution:

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

Problem 10

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

Solution:

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

📘 End of Exam-Oriented Problems

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