Solve d(f(x)/g(x))=g(x)*df(x)-f(x)*dg(x)/(g(x))^2

$$d ( \frac { f ( x ) } { g ( x ) } ) = \frac { g ( x ) \times d f ( x ) - f ( x ) \times d g ( x ) } { ( g ( x ) ) ^ { 2 } }$$

Evaluate

$$0$$

Short Solution Steps

Expand \left(gx\right)^{2}.

$$\frac{gx\frac{\mathrm{d}}{\mathrm{d}f}(x)-fx\frac{\mathrm{d}}{\mathrm{d}g}(x)}{g^{2}x^{2}}$$

Factor the expressions that are not already factored.

$$\frac{x\left(g\frac{\mathrm{d}}{\mathrm{d}f}(x)-f\frac{\mathrm{d}}{\mathrm{d}g}(x)\right)}{g^{2}x^{2}}$$

Cancel out x in both numerator and denominator.

$$\frac{g\frac{\mathrm{d}}{\mathrm{d}f}(x)-f\frac{\mathrm{d}}{\mathrm{d}g}(x)}{xg^{2}}$$

Expand the expression.

$$\frac{0}{xg^{2}}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

$$0$$