Derivatives of Algebraic Functions : Learn Formula and Proof using Solved Examples

Derivatives of Algebraic Functions are a series of formulae that can be used to differentiate algebraic functions quickly. In this article, we will learn Derivatives of Algebraic Functions and their Formulae with Proof and Solved Examples, Graphical Representation of Derivatives, and How to Find a Derivative.

An algebraic function is a function that can be defined as the root of a polynomial equation. An algebraic function uses only algebraic operations such as addition, subtraction, multiplication, and division, as well as fractional or rational exponents. Derivatives of Algebraic Functions are standard functions that give derivatives of different algebraic functions. Derivatives of Algebraic functions are used to find out solutions to differential equations.

​᠎​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 

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 or . In addition, an exponential function possesses a constant as its base and a variable as its exponent. An exp function in mathematics is expressed as , where “b” stands for the variable and “x” denotes the constant which is also termed the base of the function. The Logarithmic function is the inverse of the exponential function. Logarithmic functions in mathematics are an operator which will help you exactly calculate the exponent that will satisfy the exponential equation. The logarithmic function equation is as shown, for a>0 such that b>0 and .

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 in mathematics. The derivative of a raised to the power of x with respect to x is written in the following form in calculus.

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.

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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 Algebric Functions

Here are some solved example on Derivatives of Algebric Functions.

Solved Example:

Solution:

Using Derivatives of Exponential Functions of x by Power Rule, we get

Solved Example:

Solution:

Using Derivative of a Constant Rule, we get

Solved Example:

Solution:

Using Derivatives of Exponential Functions of x by Power Rule, we get

Solved Example: \(f(x)=\sqrt[3]{x}–\frac{1}{\sqrt{x}}[\latex]

Solution:

Here we use both Derivatives of Exponential Functions of x by Power Rule and Derivative of Square root

Solved Example: \(\)f(x)= e^{1 + x}\)

Solution:

Using Derivatives of Exponential Functions of e by Power Rule, we get

\({d\over{dx}}e^x=e^x\)

\(\begin{matrix}
\dfrac{d}{dx}\left(e^{1 + x} \right) &= \dfrac{d}{dx}\left(e \cdot e^x \right) \\
= e \dfrac{d}{dx}\left(e^x \right) \\
= e \left(e^x \right)
= e^{x+1}
\end{matrix}
\)

Solved Example: \( f(x) = \sqrt{x}\left(x^2 – 8 + \dfrac{1}{x} \right)\)

Solution: \( f(x) = \sqrt{x}\left(x^2 – 8 + \dfrac{1}{x} \right)\)

\(\begin{matrix}
\sqrt{x}\left(x^2 – 8 + \frac{1}{x} \right) &= x^{\frac{1}{2}}x^2 – 8x^{\frac{1}{2}} + x^{\frac{1}{2}}x^{-1} \\
&= x^{\frac{5}{2}}\, – 8x^{\frac{1}{2}}\, + x^{-\frac{1}{2}}
\end{matrix}\)

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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