Solve f^prime(x)=(ax-e^x)(x-2)/x^3

$$f ^ { \prime } ( x ) = \frac { ( a x - e ^ { x } ) ( x - 2 ) } { x ^ { 3 } }$$

Solve for a

$$\left\{\begin{matrix}a=\frac{e^{x}}{x}\text{, }&x\neq 2\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&x=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xx^{3}=\left(ax-e^{x}\right)\left(x-2\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{4}=\left(ax-e^{x}\right)\left(x-2\right)$$

Use the distributive property to multiply ax-e^{x} by x-2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{4}=\left(ax-e^{x}\right)x-2\left(ax-e^{x}\right)$$

Swap sides so that all variable terms are on the left hand side.

$$\left(ax-e^{x}\right)x-2\left(ax-e^{x}\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{4}$$

Use the distributive property to multiply ax-e^{x} by x.

$$ax^{2}-e^{x}x-2\left(ax-e^{x}\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{4}$$

Use the distributive property to multiply -2 by ax-e^{x}.

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

Add e^{x}x to both sides.

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

Subtract 2e^{x} from both sides.

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

Combine all terms containing a.

$$\left(x^{2}-2x\right)a=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{4}+e^{x}x-2e^{x}$$

The equation is in standard form.

$$\left(x^{2}-2x\right)a=xe^{x}-2e^{x}$$

Divide both sides by x^{2}-2x.

$$\frac{\left(x^{2}-2x\right)a}{x^{2}-2x}=\frac{\left(x-2\right)e^{x}}{x^{2}-2x}$$

Dividing by x^{2}-2x undoes the multiplication by x^{2}-2x.

$$a=\frac{\left(x-2\right)e^{x}}{x^{2}-2x}$$

Divide \left(-2+x\right)e^{x} by x^{2}-2x.

$$a=\frac{e^{x}}{x}$$

Solve for f

$$f\in \mathrm{R}$$

$$x\neq 0\text{ and }-ax^{2}+xe^{x}+2ax-2e^{x}=0$$