Solve g ( x ) = ∫ -1/x^2*f^prime(1/x) wrt x

$$g ( x ) = \int - \frac { 1 } { x ^ { 2 } } \cdot f ^ { \prime } ( \frac { 1 } { x } ) d x$$

Solve for g

$$g=-\frac{С}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Express \left(-\frac{1}{x^{2}}\right)\frac{\mathrm{d}}{\mathrm{d}x}(f) as a single fraction.

$$gx=\int \frac{-\frac{\mathrm{d}}{\mathrm{d}x}(f)}{x^{2}}\times \frac{1}{x}\mathrm{d}x$$

Multiply \frac{-\frac{\mathrm{d}}{\mathrm{d}x}(f)}{x^{2}} times \frac{1}{x} by multiplying numerator times numerator and denominator times denominator.

$$gx=\int \frac{-\frac{\mathrm{d}}{\mathrm{d}x}(f)}{x^{2}x}\mathrm{d}x$$

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

$$gx=\int \frac{-\frac{\mathrm{d}}{\mathrm{d}x}(f)}{x^{3}}\mathrm{d}x$$

The equation is in standard form.

$$xg=С$$

Divide both sides by x.

$$\frac{xg}{x}=\frac{С}{x}$$

Dividing by x undoes the multiplication by x.

$$g=\frac{С}{x}$$

Solve for f

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

$$g=-\frac{С}{x}\text{ and }x\neq 0$$