Solve 4x/(sqrt{1-x^2-1/x^2+1)*(x^2+1)^2}

$$\frac { 4 x } { ( \sqrt { 1 } - \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 } ) \cdot ( x ^ { 2 } + 1 ) ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Calculate the square root of 1 and get 1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+1}{x^{2}+1}.

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

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

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

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

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

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

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

Express \frac{2}{x^{2}+1}\left(x^{2}+1\right)^{2} as a single fraction.

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

Cancel out x^{2}+1 in both numerator and denominator.

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

Cancel out 2 in both numerator and denominator.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Calculate the square root of 1 and get 1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+1}{x^{2}+1}.

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

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

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

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

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

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

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

Express \frac{2}{x^{2}+1}\left(x^{2}+1\right)^{2} as a single fraction.

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

Cancel out x^{2}+1 in both numerator and denominator.

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

Cancel out 2 in both numerator and denominator.

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

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

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

Combine like terms.

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

Subtract 4 from 2.

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

Factor out 2.

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

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

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