Solve f^prime(x)=2ax-1/x^2

$$f ^ { \prime } ( x ) = 2 a x - \frac { 1 } { x ^ { 2 } }$$

Solve for a

$$a=\frac{1}{2x^{3}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xx^{2}=2axx^{2}-1$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{3}=2axx^{2}-1$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{3}=2ax^{3}-1$$

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

$$2ax^{3}-1=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{3}$$

Add 1 to both sides.

$$2ax^{3}=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{3}+1$$

The equation is in standard form.

$$2x^{3}a=1$$

Divide both sides by 2x^{3}.

$$\frac{2x^{3}a}{2x^{3}}=\frac{1}{2x^{3}}$$

Dividing by 2x^{3} undoes the multiplication by 2x^{3}.

$$a=\frac{1}{2x^{3}}$$

Solve for f

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

$$x=\frac{1}{\sqrt[3]{2a}}\text{ and }a\neq 0$$