# Derivatives of Exponential Functions Question

• TommyLF
In summary, the problem is to find the derivative of the function y = (6^3x-7) / (x^2-x). The solution involves using the quotient rule and recognizing that 6^{3x-7} can be rewritten as e^{(3x-7)*ln(6)}. The final result is (x^2-x)(e^{3x-7}(ln 6))(3ln 6) + (6^3x-7)(2x-1) / (x^2-x)^2. The original poster also asks for tips on how to format equations neatly using a mathematical markup language called LaTeX.
TommyLF
The problem is:

y= 6^3x-7
--------
(x^2)-x
Read: Y equals 6 to the 3x minus 7 OVER/Divided by x to the 2nd minus x

Now I thought I had to do a quotient rule there and start by doing:

(x^2-x)(e^(ln6)(3x-7))(3) and then so on from there. I just don't know what to do with that 6 to the 3x-7. Is it something to do with "e"?

$$6^{3x-7} = e^{(3x-7)*ln(6)}$$ You can see this by rewriting it as $$(e^{ln(6)})^{3x-7}$$ although the first form is easier to differentiate. So the derivative, by the chain rule, is $$e^{3x-7)*ln(6)}*ln(6)*3$$ or $$6^{3x-7}*3ln(6)$$

So would the quotient rule look like:

(x^2-x)(e^(3x-7)(ln 6))(3ln 6) + (6^3x-7)(2x-1)
--------------------------------------------
(x^2-x)^2

By the way, how do I get my question to look like nice and neat without a bunch of ^ like you just did?

TommyLF said:
By the way, how do I get my question to look like nice and neat without a bunch of ^ like you just did?

Is a mathematical markup language called LaTeX.

More specifically, the derivative of ax is ln(a) ax.

If you click on something like

$$\frac{da^x}{dx}= ln(a) a^x$$
A popup box will show the code.

## 1. What is a derivative of an exponential function?

A derivative of an exponential function is a mathematical concept that represents the rate of change of the function at a specific point. It measures how much the function is increasing or decreasing at that point.

## 2. How do you find the derivative of an exponential function?

To find the derivative of an exponential function, you can use the power rule or the logarithmic differentiation method. The power rule states that the derivative of an exponential function is equal to the natural logarithm of the base multiplied by the function itself. The logarithmic differentiation method involves taking the natural logarithm of both sides of the function and then using the power rule to find the derivative.

## 3. What is the derivative of e^x?

The derivative of e^x is equal to e^x. This can be derived using the power rule, as the natural logarithm of e is equal to 1.

## 4. Can you give an example of finding the derivative of an exponential function?

Sure, let's find the derivative of y = 3^x. Using the power rule, we can rewrite this as y = e^(xln3). Taking the derivative, we get y' = e^(xln3) * ln3 = 3^x * ln3. So, the derivative of y = 3^x is equal to 3^x multiplied by the natural logarithm of 3.

## 5. What is the significance of derivatives of exponential functions?

Derivatives of exponential functions are important in understanding the behavior of exponential growth and decay situations. They can help us determine the rate of change, maximum or minimum values, and points of inflection for these types of functions. They are also useful in many applications in science, economics, and engineering.

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