Differentiation of Algebraic Functions

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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
5 years, 7 months ago

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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, 4 months ago

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