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Derivatives of Algebraic Functions Learn Formula, Proof & Examples
IMPORTANT LINKS
Derivatives of algebraic functions help us find the rate at which a function is changing. These are a set of rules or formulas that make it easier and faster to differentiate (or find the derivative of) algebraic expressions.
An algebraic function is any function that involves operations like addition, subtraction, multiplication, division, and powers or roots (like square roots or cube roots) of variables. These functions come from solving polynomial equations.
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By using derivatives of algebraic functions, we can solve problems in calculus and also find solutions to many types of differential equations. In this topic, we also learn how to use formulas, solve problems step by step, and sometimes even look at graphs to understand how the derivative behaves.
Sum Rule
The Sum rule says the derivative of a sum of functions is the sum of their derivatives. The Difference rule says the derivative of a difference of functions is the difference of their derivatives.
Difference Rule
The Difference rule says the derivative of a difference of functions is the difference of their derivatives.
Product Rule
Sometimes we are given functions that are actually products of other functions. This means, two functions multiplied together. A special rule, the product rule, exists for differentiating products of two (or more) functions.
If y = uv then
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Quotient Rule
A special rule, the quotient rule, exists for differentiating quotients of two functions. Functions often come as quotients, by which we mean one function divided by another function. There is a formula we can use to differentiate a quotient – it is called the quotient rule.
If f and g are both differentiable, then:
Derivatives of Other Functions
The exponential function is a mathematical function denoted by
Derivatives of Exponential Functions
Let’s see how we can calculate the derivative of exponential functions.
Derivatives of Exponential Functions of x by Power Rule
The derivative of an exponential function, which contains a variable as a base and a constant as power, is called the constant power derivative rule.
x and n are literals and they represent a variable and a constant. They form an exponential term x^n. The derivative of x is raised to the power n is written in mathematical form as follows.
Derivatives of Exponential Functions of a by Power Rule
The derivative of an exponential term, which contains a variable as a base and a constant as power, is called the constant power derivative rule.
Suppose a and x represent a constant and a variable respectively then the exponential function is written as
Derivatives of Exponential Functions of e by Power Rule
The derivative of an exponential term, which contains a variable as a base and a constant as power, is called the constant power derivative rule.
Assume that x is a variable, then the natural exponential function is written as ex in mathematical form. The derivative of the ex function with respect to x is written in the following mathematical form.
Derivative of Logarithmic Function
The derivative of a logarithmic function of the variable with respect to itself is equal to its reciprocal.
Derivatives of logarithmic functions are used to find out solutions to differential equations.
Derivative of Square Root
The derivative of the square root of x can be found using the power rule. Here the power of x is ½. The derivative of x is raised to the power n is written in mathematical form as follows.
Solved Examples based on Derivatives of Algebraic Functions
Here are some solved examples on Derivatives of Algebraic Functions.
Example 1:
Solution:
Using Derivatives of Exponential Functions of x by Power Rule, we get
Example 2:
Solution:
Using Derivative of a Constant Rule, we get
\(\begin{matrix}
\dfrac{d}{dx}(2\pi) &= \dfrac{d}{dx}(\text{constant}) \\
= 0 \
\end{matrix}
\)
Example 3:
Solution:
Using Derivatives of Exponential Functions of x by Power Rule, we get
\(\begin{matrix}
\dfrac{d}{dx}\left( \frac{2}{3}x^9\right) &= \frac{2}{3} \dfrac{d}{dx}\left(x^9 \right) \\
= \frac{2}{3}\left(9 x^{9-1} \right) \\
= \frac{2}{3}(9) \left(x^8 \right) \\
= 6x^8
\end{matrix}\)
Example 4: Find the derivative of the function
f(x) = ∛x − 1/√x
Solution:
First, rewrite the function using exponents:
f(x) = x^(1/3) − x^(-1/2)
Now differentiate using the power rule:
d/dx (x^(1/3)) = (1/3) x^(-2/3)
d/dx (-x^(-1/2)) = (1/2) x^(-3/2)
Answer: f '(x) = (1/3)x^(-2/3) + (1/2)x^(-3/2).
Example 5: Find the derivative of f(x) = e^(1 + x)
Solution:
We use the rule:
d/dx (e^u) = e^u · (du/dx)
Here, u = 1 + x, so du/dx = 1
Therefore,
d/dx (e^(1 + x)) = e^(1 + x) · 1 = e^(1 + x)
Answer: f '(x) = e^(1 + x)
Example 6:
Solution:
\(\begin{matrix}
\sqrt{x}\left(x^2 – 8 + \frac{1}{x} \right) &= x^{\frac{1}{2}}x^2 – 8x^{\frac{1}{2}} + x^{\frac{1}{2}}x^{-1} \\
&= x^{\frac{5}{2}}\, – 8x^{\frac{1}{2}}\, + x^{-\frac{1}{2}}
\end{matrix}\)
Using Derivatives of Exponential Functions of x by Power Rule, we get
\(\begin{matrix}
\dfrac{d}{dx} \left(\sqrt{x}\left(x^2 – 8 + \frac{1}{x} \right) \right) &= \dfrac{d}{dx}\left(x^{\frac{5}{2}}\right) – 8\dfrac{d}{dx}\left(x^{\frac{1}{2}} \right) + \dfrac{d}{dx} \left( x^{-\frac{1}{2}} \right) \\
= \frac{5}{2} x^{\left(\frac{5}{2}\, – 1 \right)} – 8 \left(\frac{1}{2} x^{\left(\frac{1}{2}\, – 1 \right)} \right) + \left(-\frac{1}{2} \right)x^{\left(-\frac{1}{2}\, -1 \right)} \\
= \frac{5}{2}x^{\frac{3}{2}}\, – 4 x^{-\frac{1}{2}} \,- \frac{1}{2} x^{-\frac{3}{2}} \end{matrix}\)
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FAQs For Derivatives of Algebraic Functions
What is a derivative in algebra?
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.The derivative of a function, represented by
What are algebraic functions?
An algebraic function is a function that can be defined as the root of a polynomial equation.
What are the properties of algebraic functions?
Let f(x) and g(x) be two algebraic functions of x. These two functions will have the following properties.(f + g)(x) = f(x) + g(x)(f + g)(x) = f(x) - g(x)(f x g)(x) = f(x) x g(x)(f \ g)(x) = f(x) \ g(x) where g(x) is not equal to zero.Here’s a list of Derivatives of Algebraic Functions.
What is the derivative of f(x) = 1?
The derivative of a constant with respect to any variable is equal to zero.Let k be a constant with respect to x. The derivative of constant k with respect to x is written in the following mathematical form.
What is the derivative of f(x) = x?
The derivative of a variable with respect to the same variable is equal to one.
What rules are used in differentiating algebraic functions?
The most commonly used rules are: Power Rule Sum/Difference Rule Product Rule Quotient Rule Chain Rule
What is the derivative of a constant?
The derivative of a constant is always zero.
What is the quotient rule?
If a function is written as u(x) / v(x), then the derivative is found using the quotient rule: d/dx (u/v) = (v × du/dx − u × dv/dx) / v² This rule is used to differentiate a function where one expression is divided by another.
Do derivatives apply to fractional powers too?
Yes. The power rule works for all real exponents, including fractions and negative numbers. Example: d/dx (x^(2/3)) = (2/3) × x^(-1/3)
How are derivatives used in real life?
Derivatives help find speed, growth rates, slopes, and optimization in physics, economics, biology, and engineering.