Solve 5/x^2+3+7/2x^2+1

$$\frac { 5 } { x ^ { 2 } + 3 } + \frac { 7 } { 2 x ^ { 2 } + 1 }$$

Evaluate

$$\frac{17x^{2}+26}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+3 and 2x^{2}+1 is \left(2x^{2}+1\right)\left(x^{2}+3\right). Multiply \frac{5}{x^{2}+3} times \frac{2x^{2}+1}{2x^{2}+1}. Multiply \frac{7}{2x^{2}+1} times \frac{x^{2}+3}{x^{2}+3}.

$$\frac{5\left(2x^{2}+1\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}+\frac{7\left(x^{2}+3\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}$$

Since \frac{5\left(2x^{2}+1\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)} and \frac{7\left(x^{2}+3\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(2x^{2}+1\right)+7\left(x^{2}+3\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}$$

Do the multiplications in 5\left(2x^{2}+1\right)+7\left(x^{2}+3\right).

$$\frac{10x^{2}+5+7x^{2}+21}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}$$

Combine like terms in 10x^{2}+5+7x^{2}+21.

$$\frac{17x^{2}+26}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}$$

Expand \left(2x^{2}+1\right)\left(x^{2}+3\right).

$$\frac{17x^{2}+26}{2x^{4}+7x^{2}+3}$$

Differentiate w.r.t. x

$$-\frac{2x\left(34x^{4}+104x^{2}+131\right)}{\left(\left(2x^{2}+1\right)\left(x^{2}+3\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+3 and 2x^{2}+1 is \left(2x^{2}+1\right)\left(x^{2}+3\right). Multiply \frac{5}{x^{2}+3} times \frac{2x^{2}+1}{2x^{2}+1}. Multiply \frac{7}{2x^{2}+1} times \frac{x^{2}+3}{x^{2}+3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(2x^{2}+1\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)}+\frac{7\left(x^{2}+3\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)})$$

Since \frac{5\left(2x^{2}+1\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)} and \frac{7\left(x^{2}+3\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(2x^{2}+1\right)+7\left(x^{2}+3\right)}{\left(2x^{2}+1\right)\left(x^{2}+3\right)})$$

Do the multiplications in 5\left(2x^{2}+1\right)+7\left(x^{2}+3\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{2}+5+7x^{2}+21}{\left(2x^{2}+1\right)\left(x^{2}+3\right)})$$

Combine like terms in 10x^{2}+5+7x^{2}+21.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{17x^{2}+26}{\left(2x^{2}+1\right)\left(x^{2}+3\right)})$$

Use the distributive property to multiply 2x^{2}+1 by x^{2}+3 and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{17x^{2}+26}{2x^{4}+7x^{2}+3})$$

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.

$$\frac{\left(2x^{4}+7x^{2}+3\right)\frac{\mathrm{d}}{\mathrm{d}x}(17x^{2}+26)-\left(17x^{2}+26\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{4}+7x^{2}+3)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

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}.

$$\frac{\left(2x^{4}+7x^{2}+3\right)\times 2\times 17x^{2-1}-\left(17x^{2}+26\right)\left(4\times 2x^{4-1}+2\times 7x^{2-1}\right)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

Simplify.

$$\frac{\left(2x^{4}+7x^{2}+3\right)\times 34x^{1}-\left(17x^{2}+26\right)\left(8x^{3}+14x^{1}\right)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

Multiply 2x^{4}+7x^{2}+3 times 34x^{1}.

$$\frac{2x^{4}\times 34x^{1}+7x^{2}\times 34x^{1}+3\times 34x^{1}-\left(17x^{2}+26\right)\left(8x^{3}+14x^{1}\right)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

Multiply 17x^{2}+26 times 8x^{3}+14x^{1}.

$$\frac{2x^{4}\times 34x^{1}+7x^{2}\times 34x^{1}+3\times 34x^{1}-\left(17x^{2}\times 8x^{3}+17x^{2}\times 14x^{1}+26\times 8x^{3}+26\times 14x^{1}\right)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

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

$$\frac{2\times 34x^{4+1}+7\times 34x^{2+1}+3\times 34x^{1}-\left(17\times 8x^{2+3}+17\times 14x^{2+1}+26\times 8x^{3}+26\times 14x^{1}\right)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

Simplify.

$$\frac{68x^{5}+238x^{3}+102x^{1}-\left(136x^{5}+238x^{3}+208x^{3}+364x^{1}\right)}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

Combine like terms.

$$\frac{-68x^{5}-208x^{3}-262x^{1}}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$

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

$$\frac{-68x^{5}-208x^{3}-262x}{\left(2x^{4}+7x^{2}+3\right)^{2}}$$