Solve f(x+h)-f(x)=-h/x^2+xh

$$f ( x + h ) - f ( x ) = - \frac { h } { x ^ { 2 } + x h }$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=-\frac{1}{x\left(x+h\right)}\text{, }&x\neq -h\text{ and }x\neq 0\\f\in \mathrm{C}\text{, }&h=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x+h\right).

$$f\left(x+h\right)x\left(x+h\right)-fxx\left(x+h\right)=-h$$

Multiply x+h and x+h to get \left(x+h\right)^{2}.

$$f\left(x+h\right)^{2}x-fxx\left(x+h\right)=-h$$

Multiply x and x to get x^{2}.

$$f\left(x+h\right)^{2}x-fx^{2}\left(x+h\right)=-h$$

Use the distributive property to multiply -fx^{2} by x+h.

$$f\left(x+h\right)^{2}x-fx^{3}-fx^{2}h=-h$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+h\right)^{2}.

$$f\left(x^{2}+2xh+h^{2}\right)x-fx^{3}-fx^{2}h=-h$$

Use the distributive property to multiply f by x^{2}+2xh+h^{2}.

$$\left(fx^{2}+2fxh+fh^{2}\right)x-fx^{3}-fx^{2}h=-h$$

Use the distributive property to multiply fx^{2}+2fxh+fh^{2} by x.

$$fx^{3}+2fhx^{2}+fh^{2}x-fx^{3}-fx^{2}h=-h$$

Combine fx^{3} and -fx^{3} to get 0.

$$2fhx^{2}+fh^{2}x-fx^{2}h=-h$$

Combine 2fhx^{2} and -fx^{2}h to get fhx^{2}.

$$fhx^{2}+fh^{2}x=-h$$

Combine all terms containing f.

$$\left(hx^{2}+h^{2}x\right)f=-h$$

The equation is in standard form.

$$\left(hx^{2}+xh^{2}\right)f=-h$$

Divide both sides by hx^{2}+h^{2}x.

$$\frac{\left(hx^{2}+xh^{2}\right)f}{hx^{2}+xh^{2}}=-\frac{h}{hx^{2}+xh^{2}}$$

Dividing by hx^{2}+h^{2}x undoes the multiplication by hx^{2}+h^{2}x.

$$f=-\frac{h}{hx^{2}+xh^{2}}$$

Divide -h by hx^{2}+h^{2}x.

$$f=-\frac{1}{x\left(x+h\right)}$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{1}{x\left(x+h\right)}\text{, }&x\neq -h\text{ and }x\neq 0\\f\in \mathrm{R}\text{, }&h=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x+h\right).

$$f\left(x+h\right)x\left(x+h\right)-fxx\left(x+h\right)=-h$$

Multiply x+h and x+h to get \left(x+h\right)^{2}.

$$f\left(x+h\right)^{2}x-fxx\left(x+h\right)=-h$$

Multiply x and x to get x^{2}.

$$f\left(x+h\right)^{2}x-fx^{2}\left(x+h\right)=-h$$

Use the distributive property to multiply -fx^{2} by x+h.

$$f\left(x+h\right)^{2}x-fx^{3}-fx^{2}h=-h$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+h\right)^{2}.

$$f\left(x^{2}+2xh+h^{2}\right)x-fx^{3}-fx^{2}h=-h$$

Use the distributive property to multiply f by x^{2}+2xh+h^{2}.

$$\left(fx^{2}+2fxh+fh^{2}\right)x-fx^{3}-fx^{2}h=-h$$

Use the distributive property to multiply fx^{2}+2fxh+fh^{2} by x.

$$fx^{3}+2fhx^{2}+fh^{2}x-fx^{3}-fx^{2}h=-h$$

Combine fx^{3} and -fx^{3} to get 0.

$$2fhx^{2}+fh^{2}x-fx^{2}h=-h$$

Combine 2fhx^{2} and -fx^{2}h to get fhx^{2}.

$$fhx^{2}+fh^{2}x=-h$$

Combine all terms containing f.

$$\left(hx^{2}+h^{2}x\right)f=-h$$

The equation is in standard form.

$$\left(hx^{2}+xh^{2}\right)f=-h$$

Divide both sides by hx^{2}+h^{2}x.

$$\frac{\left(hx^{2}+xh^{2}\right)f}{hx^{2}+xh^{2}}=-\frac{h}{hx^{2}+xh^{2}}$$

Dividing by hx^{2}+h^{2}x undoes the multiplication by hx^{2}+h^{2}x.

$$f=-\frac{h}{hx^{2}+xh^{2}}$$

Divide -h by hx^{2}+h^{2}x.

$$f=-\frac{1}{x\left(x+h\right)}$$

Solve for h

$$\left\{\begin{matrix}h=0\text{, }&x\neq 0\\h=\frac{-fx^{2}-1}{fx}\text{, }&x\neq 0\text{ and }f\neq 0\end{matrix}\right.$$