Solve F(x):(x+5)/(x-2)

$$F ( x ) : ( x + 5 ) / ( x - 2 )$$

Evaluate

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

Short Solution Steps

Express \frac{\frac{Fx}{x+5}}{x-2} as a single fraction.

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

Apply the distributive property by multiplying each term of x+5 by each term of x-2.

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

Combine -2x and 5x to get 3x.

$$\frac{Fx}{x^{2}+3x-10}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Express \frac{\frac{Fx}{x+5}}{x-2} as a single fraction.

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

Apply the distributive property by multiplying each term of x+5 by each term of x-2.

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

Combine -2x and 5x to get 3x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{Fx}{x^{2}+3x-10})$$

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.

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

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}.

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

Simplify.

$$\frac{\left(x^{2}+3x^{1}-10\right)Fx^{0}-Fx^{1}\left(2x^{1}+3x^{0}\right)}{\left(x^{2}+3x^{1}-10\right)^{2}}$$

Multiply x^{2}+3x^{1}-10 times Fx^{0}.

$$\frac{x^{2}Fx^{0}+3x^{1}Fx^{0}-10Fx^{0}-Fx^{1}\left(2x^{1}+3x^{0}\right)}{\left(x^{2}+3x^{1}-10\right)^{2}}$$

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

$$\frac{x^{2}Fx^{0}+3x^{1}Fx^{0}-10Fx^{0}-\left(Fx^{1}\times 2x^{1}+Fx^{1}\times 3x^{0}\right)}{\left(x^{2}+3x^{1}-10\right)^{2}}$$

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

$$\frac{Fx^{2}+3Fx^{1}-10Fx^{0}-\left(F\times 2x^{1+1}+F\times 3x^{1}\right)}{\left(x^{2}+3x^{1}-10\right)^{2}}$$

Simplify.

$$\frac{Fx^{2}+3Fx^{1}+\left(-10F\right)x^{0}-\left(2Fx^{2}+3Fx^{1}\right)}{\left(x^{2}+3x^{1}-10\right)^{2}}$$

Combine like terms.

$$\frac{\left(-F\right)x^{2}+\left(-10F\right)x^{0}}{\left(x^{2}+3x^{1}-10\right)^{2}}$$

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

$$\frac{\left(-F\right)x^{2}+\left(-10F\right)x^{0}}{\left(x^{2}+3x-10\right)^{2}}$$

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

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

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

$$\frac{\left(-F\right)x^{2}-10F}{\left(x^{2}+3x-10\right)^{2}}$$