Solve 3/5x^2+4x-1+2/x+1

$$\frac { 3 } { 5 x ^ { 2 } + 4 x - 1 } + \frac { 2 } { x + 1 }$$

Evaluate

$$\frac{10x+1}{\left(5x-1\right)\left(x+1\right)}$$

Short Solution Steps

Factor 5x^{2}+4x-1.

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

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

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

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

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

Do the multiplications in 3+2\left(5x-1\right).

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

Combine like terms in 3+10x-2.

$$\frac{1+10x}{\left(5x-1\right)\left(x+1\right)}$$

Expand \left(5x-1\right)\left(x+1\right).

$$\frac{1+10x}{5x^{2}+4x-1}$$

Differentiate w.r.t. x

$$-\frac{2\left(25x^{2}+5x+7\right)}{\left(\left(5x-1\right)\left(x+1\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor 5x^{2}+4x-1.

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

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

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

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

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

Do the multiplications in 3+2\left(5x-1\right).

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

Combine like terms in 3+10x-2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1+10x}{\left(5x-1\right)\left(x+1\right)})$$

Use the distributive property to multiply 5x-1 by x+1 and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1+10x}{5x^{2}+4x-1})$$

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.

$$\frac{\left(5x^{2}+4x^{1}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(10x^{1}+1)-\left(10x^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{2}+4x^{1}-1)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

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}.

$$\frac{\left(5x^{2}+4x^{1}-1\right)\times 10x^{1-1}-\left(10x^{1}+1\right)\left(2\times 5x^{2-1}+4x^{1-1}\right)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

Simplify.

$$\frac{\left(5x^{2}+4x^{1}-1\right)\times 10x^{0}-\left(10x^{1}+1\right)\left(10x^{1}+4x^{0}\right)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

Multiply 5x^{2}+4x^{1}-1 times 10x^{0}.

$$\frac{5x^{2}\times 10x^{0}+4x^{1}\times 10x^{0}-10x^{0}-\left(10x^{1}+1\right)\left(10x^{1}+4x^{0}\right)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

Multiply 10x^{1}+1 times 10x^{1}+4x^{0}.

$$\frac{5x^{2}\times 10x^{0}+4x^{1}\times 10x^{0}-10x^{0}-\left(10x^{1}\times 10x^{1}+10x^{1}\times 4x^{0}+10x^{1}+4x^{0}\right)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

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

$$\frac{5\times 10x^{2}+4\times 10x^{1}-10x^{0}-\left(10\times 10x^{1+1}+10\times 4x^{1}+10x^{1}+4x^{0}\right)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

Simplify.

$$\frac{50x^{2}+40x^{1}-10x^{0}-\left(100x^{2}+40x^{1}+10x^{1}+4x^{0}\right)}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

Combine like terms.

$$\frac{-50x^{2}-10x^{1}-14x^{0}}{\left(5x^{2}+4x^{1}-1\right)^{2}}$$

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

$$\frac{-50x^{2}-10x-14x^{0}}{\left(5x^{2}+4x-1\right)^{2}}$$

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

$$\frac{-50x^{2}-10x-14}{\left(5x^{2}+4x-1\right)^{2}}$$