An algebraic function is a function that can be written using a finite number of the basic operations of arithmetic (i.e., addition, multiplication, and exponentiation). In order to take the derivative of these functions, we will need the power rule.

**The Power Rule:** The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

**Exponential Functions:** An exponential function with base $e$ is its own derivative. That is to say, if $f(x) = e^x$, then $f'(x) = e^x$ as well.

**Logarithmic Function:** Logarithmic functions with base $e$ have derivatives of the following form: if $f(x) = \log_e x$, then $f'(x) = \frac{1}{x}$.

## What is the slope of $e^x$ when $x = \ln 5$?

To find the slope, we find the derivative of $e^x$. But that is simply $e^x$ by the above rule. So the slope when $x=5$ is $e^{\ln 5} = 5$. $_\square$

## If $f(x) = x^3 - 3x^4$, what is $f'(-4)$?

Since we can take the derivative of each term separately,

$\begin{aligned} f'(x) &= 3x^{3-1} - 3(4)x^{4-1} \\ &= 3x^2-12x^3. \end{aligned}$

Evaluating, $f'(-4) = 3(-4)^2-12(-4)^3=48+768=816$. $_\square$

No vote yet

1 vote

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestIf a function is twice differentiable at a point,what will be the position of first and second derivative relative to each other?I mean geometric interpretation.

Log in to reply