Animation of Exponential function and Derivative (for Students)

This page explains the relationship between an exponential function whose base is a positive real number and its derivative, using graph animations and still images.

(There is the Japanese(日本語) page.)

(Last updated date: March 24, 2021)

Preface

Mathematical analysis books always include a graph of the exponential function ${\mathrm{e}}^x$, where $\mathrm{e}$ is Euler’s number (or Napier’s constant). The books also always include that the derivative of the expression is equal to the original expression.

However, the graph of the exponential function $a^x$, whose base is a positive real number $a$, may not be shown. Furthermore, the graph of the derivative of the expression will not be shown first.

Therefore, in order to visually understand the relationship between the exponential function $a^x$ and its derivative, I have created animations (movies) of these graphs. In addition, in order to better understand these graphs, I explain some characteristic examples of these graphs with still images. The animations and still images are published in the next chapter.

This page is intended for people who study mathematics (students, etc.), especially those who are not used to reading mathematical formulas and their graphs.

The animations and images are created with SageMath. The SageMath script is published in the next page. If you are interested, please take a look at that page as well.

Animations of exponential function and its derivative

Formulas

The formulas for the exponential function $y$ with the base $a$ (positive real number, $a\ne 1$) and its derivative $y^{\prime }$ are shown below.

$$\begin{array}{rl}y& =a^x\\ y^{\prime }& =a^x\log a\end{array}$$

The base of the logarithm used in this page is Euler’s number $\mathrm{e}$. Therefore, the above $\log a$ can be written exactly as ${\log }_{\mathrm{e}}a$. Such a logarithm is called the natural logarithm. However, in mathematics, the base $\mathrm{e}$ of the natural logarithm is often omitted, so this page will also write it as $\log a$.

The following two points should be noted here.

• Both formulas include “$a^x$”.
• Only the derivative formula includes “$\log a$”.

Since the $a^x$ part is the same, the values of both formulas vary similarly, depending on the values of $a$ and $x$. In addition, the value of the derivative formula is always the value of this $a^x$ multiplied by $\log a$.

These formulas will be explained in a little more detail later. With the above in mind, look at the following animations.

Animations

The following animation shows the variation in the graphs of the two formulas above when $a$ in the formulas is varied from $1.1$ to $5$. First, watch the animation to get a rough idea of the overall behavior.

If the above video does not display correctly, please try the links below.

• MP4 format (High 4:4:4 Predictive profile)
• MP4 format (Main profile)
• WebM format

The individual graphs in the animation show the curves of the two formulas mentioned above, the exponential function $y=a^x$ and its derivative $y^{\prime }=(a^x{)}^{\prime }=a^x\log a$. As $a$ varies, the shape of these two curves also varies.

What you should pay attention to here is the following variation when $a$ varies.

• The derivative curve gradually approaches the exponential curve.
• At one point, the two curves match.
• After that, the derivative curve gradually moves away from the exponential curve.

In comparison to the exponential curve, the derivative curve seems to “move”.

To understand the behavior of such curves, recall the “$a^x$” and “$\log a$” explained in the previous section.

The formula for the exponential function is $a^x$ itself. Therefore, the behavior of the exponential curve expresses how $a^x$ varies.

On the other hand, the formula for the derivative is $a^x$ multiplied by $\log a$. Therefore, the derivative curve is affected by the two equations $a^x$ and $\log a$. The “moving” behavior of the derivative curve can be thought of as the effect of $\log a$. Thus, the point of understanding such behavior is to understand how $\log a$ varies.

Accordingly, look at the following animation of the graph of the logarithmic function $\log x$.

If the above video does not display correctly, please try the links below.

• MP4 format (High 4:4:4 Predictive profile)
• MP4 format (Main profile)
• WebM format

The symbol “$\approx$” means “approximately equal”.

In the graph of the logarithmic function $\log x$, auxiliary lines and arrows are added to emphasize the points that I want users to pay attention to. The shape of the $\log x$ curve does not vary throughout the animation. The red dot in the graph indicates that $a$ is substituted for $x$ in $\log x$.

What you should pay attention to here is the following variation in the value of $\log a$.

• From a value smaller than $1$, it gradually becomes larger and approaches $1$.
• At one point, it becomes $1$.
• After that, it gradually becomes larger than $1$.

In this way, $\log a$ varies over “the range of numbers, including $1$”.

As mentioned earlier, the derivative curve is the product of $a^x$ multiplied by $\log a$. Therefore, the derivative curve varies as follows.

• From the product of $a^x$ multiplied by a value smaller than $1$, it gradually becomes the product multiplied by a larger value.
• At one point, it becomes the product of $a^x$ multiplied by $1$.
• After that, it gradually becomes the product of $a^x$ multiplied by a value larger than $1$.

Thus, by understanding the variation in $\log a$, one can understand the “moving” behavior of the derivative curve.

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The two animations mentioned above are better understood if you watch them at the same time. Accordingly, I have also created a new animation that consists of these two animations side by side. In this page, this new animation is named “Parallel animation”.

However, this parallel animation does not fit within the blog page due to its long width. Therefore, I have decided to display it on a separate page (the page with only the animation).

Please use the following links (select the video format that your web browser supports).

• MP4 format (High 4:4:4 Predictive profile)
• MP4 format (Main profile)
• WebM format

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I will summarize what I have explained up to this point in the form of expressions, not sentences.

When $a$ varies, the curves of the two formulas, the exponential function $y=a^x$ and its derivative $y^{\prime }=a^x\log a$, have the following relationship.

$$\begin{array}{rlrl}y& >y^{\prime }& (\log a& <1)\\ y& =y^{\prime }& (\log a& =1)\\ y& <y^{\prime }& (\log a& >1)\end{array}$$

In the next section, concrete examples of these three states will be shown with still images.

Still images

In the previous explanations, to understand the behavior of the curve, the expressions “$a^x$” and “$\log a$” are focused on, and the value of $a$ itself is not given much attention.

However, the value of $a$ is also important when understanding the exponential function. Especially important is the value of $a$ when $\log a=1$. That is Euler’s number $\mathrm{e}=2.71828\dots$.

In the animation shown earlier, the graph for $a=\mathrm{e}$ is also drawn. However, in animation, individual graphs are displayed only momentarily, so it is difficult to see this graph with the naked eye. Therefore, after this, the graph for $a=\mathrm{e}$ will be shown in a still image.

Furthermore, this $a=\mathrm{e}$ corresponds to the second of the three state expressions described at the end of the previous section. The remaining two states can also be better understood by checking the still image graphs. Thus, after this, the graphs for $a=2,4$ will also be shown in still images.

Example 1 :   $a=2$

The still images of the two graphs when $a=2$ are shown below.

For reference, here is a link to a “parallel graph” that shows the two graphs above side by side.

Parallel graph (a=2)

Be careful that $a=2$ is less than $\mathrm{e}$, $(a=2<\mathrm{e})$.

Substituting $a=2$ for $x$ in the logarithmic function $\log x$, gives the following.

$$\log a=\log 2\approx 0.693<1$$

On the graph of the logarithmic function, it can also be seen that the value on the $y$-axis of the red dot is less than $1$.

Next, substituting $a=2$ into the formulas of the exponential function and its derivative, gives the following.

$$\begin{array}{rlrl}y& =a^x& & =2^x\\ y^{\prime }& =a^x\log a& & =2^x\log 2\approx 2^x\times (0.693)\end{array}$$

$$\therefore y>y^{\prime }$$

The symbol “$\therefore$” means “therefore”.

On the graph of the exponential function, it can be also seen that the value of the exponential curve is always greater than the value of the derivative curve.

Example 2 :   $a=\mathrm{e}$

The still images of the two graphs when $a=\mathrm{e}$ are shown below.

Parallel graph (a=e)

Substituting $a=\mathrm{e}$ for $x$ in the logarithmic function $\log x$, gives the following.

$$\log a=\log \mathrm{e}=1$$

On the graph of the logarithmic function, it can also be seen that the value on the $y$-axis of the red dot is equal to $1$.

Next, substituting $a=\mathrm{e}$ into the formulas of the exponential function and its derivative, gives the following.

$$\begin{array}{rlrl}y& =a^x& & ={\mathrm{e}}^x\\ y^{\prime }& =a^x\log a& & ={\mathrm{e}}^x\log \mathrm{e}={\mathrm{e}}^x\end{array}$$

$$\therefore y=y^{\prime }$$

On the graph of the exponential function, it can be also seen that the exponential curve matches the derivative curve.

This graph is created by drawing a curve in the order of “exponential function, derivative function”. Therefore, the exponential curve is hidden under the derivative curve and cannot be seen.

What has been said here is important and is thus summarized below.

The exponential function $y=a^x$ and its derivative $y^{\prime }=a^x\log a$ match when $a=\mathrm{e}$, and become $y=y^{\prime }={\mathrm{e}}^x$.

Example 3 :   $a=4$

The still images of the two graphs when $a=4$ are shown below.

Parallel graph (a=4)

Be careful that $a=4$ is greater than $\mathrm{e}$, $(a=4>\mathrm{e})$.

Substituting $a=4$ for $x$ in the logarithmic function $\log x$, gives the following.

$$\log a=\log 4\approx 1.386>1$$

On the graph of the logarithmic function, it can also be seen that the value on the $y$-axis of the red dot is greater than $1$.

Next, substituting $a=4$ into the formulas of the exponential function and its derivative, gives the following.

$$\begin{array}{rlrl}y& =a^x& & =4^x\\ y^{\prime }& =a^x\log a& & =4^x\log 4\approx 4^x\times (1.386)\end{array}$$

$$\therefore y<y^{\prime }$$

On the graph of the exponential function, it can be also seen that the value of the exponential curve is always less than the value of the derivative curve.

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After understanding the three examples of $a=2,\mathrm{e},4$, if you look at the preceding animation again, you will be able to better understand the variations in the two curves of the exponential function and its derivative.

The chapter “Animations of exponential function and its derivative” ends here.

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Next page: SageMath Script to create Animation and Image of Exponential function