Differentiation of Algebraic Functions

This note has been used to help create the Power Rule wiki

Definition

Algebraic Functions

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)=xn f(x) = x^n , then f′(x)=nxn−1 f'(x) = nx^{n-1} .

Exponential and Logarithmic Functions

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

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

Technique

What is the slope of ex e^x when x=ln⁡5 x = \ln 5 ?

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

 

If f(x)=x3−3x4 f(x) = x^3 - 3x^4 , what is f′(−4) f'(-4) ?

Since we can take the derivative of each term separately,

f′(x)=3x3−1−3(4)x4−1=3x2−12x3. \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 f'(-4) = 3(-4)^2-12(-4)^3=48+768=816 . □ _\square

#Calculus #Differentiation #DifferentiationOfAlgebraicFunctions #KeyTechniques

Note by Arron Kau
6 years ago

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:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. list
  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

If 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.

Prabhat Saini - 5 years, 10 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...