Solve a/(x+2)*x/(2x-1)

$$\frac { a } { ( x + 2 ) } \times \frac { x } { ( 2 x - 1 ) }$$

Evaluate

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

Short Solution Steps

Multiply \frac{a}{x+2} times \frac{x}{2x-1} by multiplying numerator times numerator and denominator times denominator.

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

Apply the distributive property by multiplying each term of x+2 by each term of 2x-1.

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

Combine -x and 4x to get 3x.

$$\frac{ax}{2x^{2}+3x-2}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Multiply \frac{a}{x+2} times \frac{x}{2x-1} by multiplying numerator times numerator and denominator times denominator.

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

Apply the distributive property by multiplying each term of x+2 by each term of 2x-1.

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

Combine -x and 4x to get 3x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{ax}{2x^{2}+3x-2})$$

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.

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

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}.

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

Simplify.

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

Multiply 2x^{2}+3x^{1}-2 times ax^{0}.

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

Multiply ax^{1} times 4x^{1}+3x^{0}.

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

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

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

Simplify.

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

Combine like terms.

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

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

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

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

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

For any term t, t\times 1=t and 1t=t.

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