Solve d^-frac{1{2}}div(d^-frac{1{2}}divd^-frac{5{2}})

$$d ^ { - \frac { 1 } { 2 } } \div ( d ^ { - \frac { 1 } { 2 } } \div d ^ { - \frac { 5 } { 2 } } )$$

Differentiate w.r.t. d

$$-\frac{5}{2d^{\frac{7}{2}}}$$

Steps Using Chain Rule

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{5}{2} from -\frac{1}{2} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{d^{-\frac{1}{2}}}{d^{2}})$$

Rewrite d^{2} as d^{-\frac{1}{2}}d^{\frac{5}{2}}. Cancel out d^{-\frac{1}{2}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{1}{d^{\frac{5}{2}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(d^{\frac{5}{2}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}d}(d^{\frac{5}{2}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(d^{\frac{5}{2}}\right)^{-2}\times \frac{5}{2}d^{\frac{5}{2}-1}$$

Simplify.

$$-\frac{5}{2}d^{\frac{3}{2}}\left(d^{\frac{5}{2}}\right)^{-2}$$

Evaluate

$$\frac{1}{d^{\frac{5}{2}}}$$