Solve - 1 implify : 1/(1 + x + x ^ 2) - 1/(1 - x + x ^ 2) - (2x)/(1 + x ^ 2 + x ^ 4) 33 54 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac { 1 } { 1 + x + x ^ { 2 } } - \frac { 1 } { 1 - x + x ^ { 2 } } - \frac { 2 x } { 1 + x ^ { 2 } + x ^ { 4 } }$$

Evaluate

$-\frac{4x}{x^{4}+x^{2}+1}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $1+x+x^{2}$ and $1-x+x^{2}$ is $\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)$. Multiply $\frac{1}{1+x+x^{2}}$ times $\frac{x^{2}-x+1}{x^{2}-x+1}$. Multiply $\frac{1}{1-x+x^{2}}$ times $\frac{x^{2}+x+1}{x^{2}+x+1}$.

$$\frac{x^{2}-x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{x^{2}+x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}}$$

Since $\frac{x^{2}-x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ and $\frac{x^{2}+x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-x+1-\left(x^{2}+x+1\right)}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}}$$

Do the multiplications in $x^{2}-x+1-\left(x^{2}+x+1\right)$.

$$\frac{x^{2}-x+1-x^{2}-x-1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}}$$

Combine like terms in $x^{2}-x+1-x^{2}-x-1$.

$$\frac{-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}}$$

Factor $1+x^{2}+x^{4}$.

$$\frac{-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$$

Since $\frac{-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ and $\frac{2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$$

Combine like terms in $-2x-2x$.

$$\frac{-4x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$$

Expand $\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)$.

$$\frac{-4x}{x^{4}+x^{2}+1}$$

Differentiate w.r.t. x

$\frac{4\left(3x^{4}+x^{2}-1\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $1+x+x^{2}$ and $1-x+x^{2}$ is $\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)$. Multiply $\frac{1}{1+x+x^{2}}$ times $\frac{x^{2}-x+1}{x^{2}-x+1}$. Multiply $\frac{1}{1-x+x^{2}}$ times $\frac{x^{2}+x+1}{x^{2}+x+1}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}-x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{x^{2}+x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}})$$

Since $\frac{x^{2}-x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ and $\frac{x^{2}+x+1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}-x+1-\left(x^{2}+x+1\right)}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}})$$

Do the multiplications in $x^{2}-x+1-\left(x^{2}+x+1\right)$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}-x+1-x^{2}-x-1}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}})$$

Combine like terms in $x^{2}-x+1-x^{2}-x-1$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{1+x^{2}+x^{4}})$$

Factor $1+x^{2}+x^{4}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}-\frac{2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)})$$

Since $\frac{-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ and $\frac{2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x-2x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)})$$

Combine like terms in $-2x-2x$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-4x}{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)})$$

Use the distributive property to multiply $x^{2}+x+1$ by $x^{2}-x+1$ and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-4x}{x^{4}+x^{2}+1})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(x^{4}+x^{2}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(-4x^{1})-\left(-4x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+x^{2}+1)\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is $0$. The derivative of $ax^{n}$ is $nax^{n-1}$.

$$\frac{\left(x^{4}+x^{2}+1\right)\left(-4\right)x^{1-1}-\left(-4x^{1}\left(4x^{4-1}+2x^{2-1}\right)\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

Simplify.

$$\frac{\left(x^{4}+x^{2}+1\right)\left(-4\right)x^{0}-\left(-4x^{1}\left(4x^{3}+2x^{1}\right)\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

Multiply $x^{4}+x^{2}+1$ times $-4x^{0}$.

$$\frac{x^{4}\left(-4\right)x^{0}+x^{2}\left(-4\right)x^{0}-4x^{0}-\left(-4x^{1}\left(4x^{3}+2x^{1}\right)\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

Multiply $-4x^{1}$ times $4x^{3}+2x^{1}$.

$$\frac{x^{4}\left(-4\right)x^{0}+x^{2}\left(-4\right)x^{0}-4x^{0}-\left(-4x^{1}\times 4x^{3}-4x^{1}\times 2x^{1}\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{-4x^{4}-4x^{2}-4x^{0}-\left(-4\times 4x^{1+3}-4\times 2x^{1+1}\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

Simplify.

$$\frac{-4x^{4}-4x^{2}-4x^{0}-\left(-16x^{4}-8x^{2}\right)}{\left(x^{4}+x^{2}+1\right)^{2}}$$

Combine like terms.

$$\frac{12x^{4}+4x^{2}-4x^{0}}{\left(x^{4}+x^{2}+1\right)^{2}}$$

For any term $t$ except $0$, $t^{0}=1$.

$$\frac{12x^{4}+4x^{2}-4}{\left(x^{4}+x^{2}+1\right)^{2}}$$