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