# Derivative of an exponential function

• magma_saber
In summary, the derivative of e^(x^3) is 3x^2e^(x^3), and the derivative of (ln(1/x))^2 is 2ln(1/x)(-1/x^2) or 2ln(1/x)(-1/x^2) with the correct use of the chain rule.
magma_saber

## Homework Statement

What is the derivative of ex3? also what is the derivative of (ln1/x)2

## The Attempt at a Solution

is it 3x2e3x2?
2(ln1/x)(x)?

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Your solution for d(e^(x^3))/dx is incorrect. Remember d(e^u)/dx = u'e^u where u is a function of x. For the other derivative, let u = ln(1/x) and then apply the chain rule.

No. No.
Both problems require the chain rule.
For the second problem, d/dx((ln(1/x))^2) = 2* ln(1/x) * d/dx(1/x). You made a mistake in your derivative of 1/x.

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so the second is 2(ln1/x)(-x^-2)?
and the first one is x^3e^x^3?

Right for the second one, but I would write it as 2 ln(1/x) (-1/x^2) or even better as
$$2 ln(\frac{1}{x}) \frac{-1}{x^2}$$
Not right for the first. As jgens said, d/dx(e^u) = du/dx * e^u (where u = x^3).

Your first derivative is still incorrect. d(e^u)/dx = u'e^u, not ue^u.
Your second derivative is also incorrect. Mark gave you an incorrect expression to differentiate. You need to find 2*ln(x)*d(ln(1/x))/dx which also requires the chain rule.

Mark44 said:
Right for the second one, but I would write it as 2 ln(1/x) (-1/x^2) or even better as
$$2 ln(\frac{1}{x}) \frac{-1}{x^2}$$

Don't forget d(ln(1/x))/dx, which certainly is not -1/x^2!

so its 3x2ex3?

Correct.

d(ln1/x) is x^2?

No, let u = 1/x then you need to find d(lnu)/dx which will be (1/u)(du/dx).

Might be easier if you just look at it as:

$$\frac{d}{dx}\;(-ln(x))$$

Then it's just straight forwardly obvious.

That certainly should make it simpler for the original poster!

## 1. What is the general formula for finding the derivative of an exponential function?

The general formula for finding the derivative of an exponential function is y = ab^x, where a is the base and b is the constant. The derivative of this function is equal to the natural logarithm of the base, multiplied by the original function. In mathematical notation, this can be written as dy/dx = ln(b) * ab^x.

## 2. How do you use the chain rule to find the derivative of an exponential function?

To use the chain rule to find the derivative of an exponential function, you first need to rewrite the function in the form y = e^u, where u is the exponent. Then, you can apply the chain rule, which states that the derivative of y with respect to x is equal to the derivative of u with respect to x, multiplied by e^u. In other words, dy/dx = du/dx * e^u. Finally, substitute the original exponent back in for u to get the final result.

## 3. Can the derivative of an exponential function be negative?

Yes, the derivative of an exponential function can be negative. This will occur when the base of the exponential function is a fraction less than 1. In this case, the derivative will always be negative, indicating that the function is decreasing as x increases.

## 4. How does the derivative of an exponential function relate to its graph?

The derivative of an exponential function is directly related to the slope of the function's graph at any point. Since the derivative represents the rate of change of the function, a positive derivative indicates a positive slope, meaning the function is increasing. Similarly, a negative derivative indicates a negative slope, meaning the function is decreasing. The derivative also determines the concavity of the function's graph, with a positive derivative indicating a concave up graph and a negative derivative indicating a concave down graph.

## 5. Is the derivative of an exponential function always an exponential function?

No, the derivative of an exponential function is not always an exponential function. In some cases, the derivative may be a polynomial, trigonometric function, or another type of function. The derivative will retain some characteristics of the original exponential function, such as having an exponential term in its equation, but it may also have additional terms depending on the specific function and its variables.

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