The derivative of e x is quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, e x !
What does this mean? It means the slope is the same as the function value (the y -value) for all points on the graph.
Example: Let's take the example when x = 2. At this point, the y -value is e 2 ≈ 7.39.
Since the derivative of e x is e x. then the slope of the tangent line at x = 2 is also e 2 ≈ 7.39.
We can see that it is true on the graph:
Let's now see if it is true at some other values of x .
We can see that at x = 4, the y -value is 54.6 and the slope of the tangent (in red) is also 54.6.
At x = 5, the y -value is 148.4, as is the value of the derivative and the slope of the tangent (in green).Other Formulas for Derivatives of Exponential Functions
If u is a function of x. we can obtain the derivative of an expression in the form e u :
If we have an exponential function with some base b. we have the following derivative:
[These formulas are derived using first principles concepts. See the chapter on Exponential and Logarithmic Functions if you need a refresher on exponential functions before starting this section.]Example 1
Find the derivative of y = 10 3x.Example 2
The derivative of an exponential function.The derivative of the exponential function with base 2
In order to take the derivative of the exponential function, say \begin
Did we make any progress? We calculated that the derivative of the function $f(x)=2^x$ is equal to the function itself ($2^x$) times a mysterious looking expression involving a limit. But, the mysterious expression involving a limit is just a single number. Nothing more, nothing less. What number is it? It's an irrational number, but you can estimate it yourself by calculating $(2^h-1)/h$ for smaller and smaller values of $h$.Use the following applet to estimate $$\lim_
Use the following applet to convince yourself that the result is valid. Check that the derivative $f'(x)$ is really a multiple of the function $f(x)$ by verifying that the ratio $f'(x)/f(x)$ doesn't change as you change $x$. This ratio should be $f'(0)$, which should be the mysterious number you estimated to four digits as well as the slope of the tangent line when $x=0$.
The derivative of an exponential function. Illustration of how the derivative of the exponential function is a multiple of the function, where that multiple is the derivative at zero. The graph of the function $f(x)=b^x$, where you can enter a value for $b$, is shown by the thick blue curve. The graph of its derivative $f'(x)$ is shown by the thin green curve, and the ratio $f'(x)/f(x)$ is shown by the horizontal gray line. The value of the function and its derivative evaluated at $x_0$ are displayed at the left and illustrated by the blue and green points on the curves. You can change $x_0$ by typing in a new value or dragging one of the points. The red line is the tangent line at $x_0$ with slope $f'(x_0)$. The ratio $f'/f$ evaluate at $x_0$ is also displayed at the left. If you check the “more options” check box, the function changes to $f(x)=c b^
Does the result that we obtained for $f(x)=2^x$ generalize to other bases besides base 2? See if you can get similar results for the exponential function $$g(x)=b^x$$ where the base $b$ is a positive parameter.First, repeat the same analytic calculation, starting with the limit definition and replacing 2 with $b$. You should obtain that \begin
Second, pick two other values of $b$ besides $b=2$. Try one value of $b$ less than one and one value of $b$ greater than one, so you experiment with both exponential decay and exponential growth. For each of your choices for $b$, use the first applet to estimate the first four digits of the value of the mysterious limit expression defining $g'(0)$. Then, use the second applet to verify that this value is both the derivative $g'(0)$ at zero and the ratio between the derivative $g'(x)$ and the function $g(x)$.The exponential function with base $e$ As you should have experienced, the mysterious limit expression $$\lim_
Another way to see this is to plot the mysterious limit as a function of $b$ and to observe that it crosses one when $b=e$.
We can conclude that for $b=e$, the derivative of the function $$g(x)=e^x,$$ evaluated at zero is $g'(0)=1$. This means that the derivative of $g$ is exactly the function itself: \begin
This fundamental property of $e^x$, that it is its own derivative, is one of the reasons that $e$ is the most common base used for the exponential function throughout the sciences. The function $e^x$ is often referred to as simply the exponential function .Adding parameters to the exponential function
When discussing the exponential function. we introduced a few parameters, writing the general exponential function as $$f(x)=cb^
Do the parameters $c$ and $k$ change the properties of the exponential function? So far, we've just looked at the case where $c=1$ and $k=1$ and found that $f'(x)=f(x)$. Is $f'(x)=f(x)$ for other values of $c$ and $k$? You can use the second applet. above, to explore this question. Check the “more options” check box, which will reveal boxes in which to change $c$ and $k$. You can observe if the graph of $f$ and $f'$ are still identical and if the ratio $f'/f$ is still one when $b=e$ but $c$ and $k$ are different values. If the ratio $f'(x_0)/f(x_0)$ changes when you change $c$ or $k$, is the value still independent of $x_0$? (In other words, is $f'$ still a multiple of $f$?) How does the ratio depend on $c$ and $k$?To obtain a definitive answer on the derivative of $f(x)=ce^
Except for the factor of $k$ in the exponent, we have exactly the same expression as before. The parameter $c$ doesn't change the result (as presumably you determined from exploring the applet). The parameter $k$ does change things, though, as it alters the mysterious limit expression. Can we reduce this new mysterious limit back into the old one?We can get our old mysterious limit back with just one manipulation. Let's define a new quantity to be equal to the exponent in our new expression: $w = kh$. Since $h=w/k$, we can rewrite the fraction inside the limit as $$\frac
Let's write this result using some fancy notation. Recall that we can write the derivative $f'(x)$ also as $\diff
The answer is one simple trick. We can turn any exponential base $b$ into the nicer exponential base $e$. We just have to remember that the inverse of the exponential function $e^x$ is the logarithm base $e$, or the natural logarithm. We often write the natural logarithm of $x$ simply as $\log x$, because if you are going to take a logarithm and no one tells you otherwise, you might as well do the natural thing and use base $e$. To be absolutely explicit, we can also use the notation $\ln x$ for the natural logarithm, which we'll use for now.
Since $\ln x$ and $e^x$ are inverses, it follows that for any number $x>0$, we can also write $x$ as $$x = e^<\ln x>.$$ If instead of $x$, we insert the expression $b^x$, we can rewrite $b^x$ as $$b^x = e^<\ln b^x> = e^
Given that $k=\ln b$, we can use formula \eqref
Limit of b to the h minus one over h as h tends to zero converges to the natural logarithm. Demonstration that a function $m(b)=(b^h-1)/h$ approaches the natural logarithm in the limit that the parameter h goes to zero. For a given value of $h$, determined by the red slider, $(b^h-1)/h$ is plotted as a function of $b$ by the thin green curve. The graph of the function $\ln b$ is plotted by the thick blue curve. As you change $h$ to make it closer and closer to zero, the thin green curve converges to the thick blue curve, demonstrating that $(b^h-1)/h$ approaches $\ln b$ as $h$ approaches zero.
It's a lot nicer to have a formula for the derivative of $b^x$ without the mysterious limit, using $\ln b$ instead. Usually, in calculus, though, we'll stay away from bases another than the natural one $e$. The most important of the formulas that we've derived are $$\diff<>
In fact, $e^
The worksheet on exploring the derivative of the exponential function contains questions to guide you through discovering the properties of the exponential function derivative.
The following problems involve the integration of exponential functions. We will assume knowledge of the following well-known differentiation formulas.
where a is any positive constant not equal to 1 and is the natural (base e ) logarithm of a. These formulas lead immediately to the following indefinite integrals.
As you do the following problems, remember these three general rules for integration.
where n is any constant not equal to -1,
where k is any constant, and
Because the integral
where k is any nonzero constant, appears so often in the following set of problems, we will find a formula for it now using u-substitution so that we don't have to do this simple process each time. Begin by letting
Now substitute into the original problem, replacing all forms of x. and getting
We now have the following variation of formula 1.).
The following often-forgotten, misused, and unpopular rules for exponents will also be helpful.Most of the following problems are average. A few are challenging. Knowledge of the method of u-substitution will be required on many of the problems. Make precise use of the differential notation dx and du and always be careful when arithmetically and algebraically simplifying expressions.
Click HERE to see a detailed solution to problem 1.
Click HERE to see a detailed solution to problem 2.
Click HERE to see a detailed solution to problem 3.
Click HERE to see a detailed solution to problem 4.
Click HERE to see a detailed solution to problem 5.
Click HERE to see a detailed solution to problem 6.
Click HERE to see a detailed solution to problem 7.
Click HERE to see a detailed solution to problem 8.
Click HERE to see a detailed solution to problem 9.
Click HERE to see a detailed solution to problem 10.
Click HERE to see a detailed solution to problem 11.
Click HERE to see a detailed solution to problem 12.
Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address.
So if I use the chain rule to take the partial derivative with respect to t2 then I hold t1 to be a constant. I get the result:
Is this correct for the partial of the function with respect to t2
Now for the second partial: take partial of this with respect to t2 of
I tried to use your Latex generated formulas to write out my results. I hope this is the correct way to do it. I saw the tutorial on Latex on the website but am not sure how to input into my messages. Do you go offsite to do that or do you put the equations directly into the message using the Latex code?
I type them directly into the box. Don't worry, I can see what you typed by clicking the 'quote' button. I'll respond again in a moment after I've reviewed what you did.
Here is a section where you can ask for latex help if you'de like to learn how to do it. The problem you're working on right now is very difficult to type, so I'm not surprised you made so many latex errors. I made several errors myself when I was typing the derivatives. The exponents inside exponents make it confusing.
Below is an essay on "4.05 Graphing Exponential Functions" from Anti Essays, your source for research papers, essays, and term paper examples.
04.05 Graphing Exponential Function
Data is everywhere. No matter where you go, thanks to new technology, data is accessible at your fingertips. However, to make data understandable, it needs to be visualized.
* Gordon is a financial analyst. He looks at tables and charts of data all day long. When he wants to present his findings, he uses graphs.
* Julie is a biologist. She collects data on the size and population of various species. When she wants to explain the data, she uses graphs.
* Julian is a video game designer. He plans the points needed to move to new levels within the game. When he wants to demonstrate the growth, he uses graphs.
Graphing is a fundamental way to present data because it can help make predictions, assist in comparing different functions, and demonstrate how changes can affect expected outcomes.
Take a look at some graphs of functions with exponents!
This function is f(x)=x2.
This function is f(x)=x
. which is the same as f(x)=√x.
This function is f(x)=x
. which is the same as f(x)=
This is the function f(x)=2x.
These are the types of functions this lesson will focus on.
After completing this lesson, you will be able to answer these questions:
* How can graphs be created from verbal expressions?
* How can the key features of an exponential function be compared through tables and graphs?
* What effect will introducing a constant have on an exponential function?
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Can anyone present an intuitive reason for why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves?
i.e. $$ y = e^
You must consider that there are some exponential functions such as $1.00001^x$ that clearly grows much slower than itself and there are functions such as $(100^<100>)^x$ or even $10^x$ that clearly grow faster than itself. This can be determined by looking at a graph or by doing some numerical calculations.Now consider the derivative of $a^x$. This is equal to $$\lim_
And $\log(x) = \log_e(x)$. The existence of the value $e$ can be justified because one can graphically determine that $1.00001^x$ grows slower than itself and $10^x$ grows faster as mentioned before. That means there should be an "$e$", $0 \lt e \lt 10$.
Polynomial functions can grow faster than themselves on an interval but as $x \to \infty$ the polynomial with the higher degree will be larger in magnitude for any polynomial. This is why this result does not hold for polynomials as well; the derivative of a polynomial has a degree of one less than the polynomial itself.
answered Mar 10 '14 at 20:03
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B.1.6 – DERIVATIVES OF EXPONENTIAL FUNCTIONS
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B.1.6 – DERIVATIVES OF EXPONENTIAL FUNCTIONS
(A) EXPLORATION – PART 1
(B) EXPLORATION – PART 2
EXPLORATION – PART 3
EXPLORATION – PART 4