Solve (frac{c^{4}d^3}{cd^2})(d^2/c^3)

$$( \frac { c ^ { 4 } d ^ { 3 } } { c d ^ { 2 } } ) ( \frac { d ^ { 2 } } { c ^ { 3 } } )$$

Evaluate

$$d^{3}$$

Solution Steps

Cancel out cd^{2} in both numerator and denominator.

$$dc^{3}\times \frac{d^{2}}{c^{3}}$$

Express d\times \frac{d^{2}}{c^{3}} as a single fraction.

$$\frac{dd^{2}}{c^{3}}c^{3}$$

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

$$\frac{d^{3}}{c^{3}}c^{3}$$

Cancel out c^{3} and c^{3}.

$$d^{3}$$

Differentiate w.r.t. d

$$3d^{2}$$

Steps Using Definition of a Derivative

Cancel out cd^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}d}(dc^{3}\times \frac{d^{2}}{c^{3}})$$

Express d\times \frac{d^{2}}{c^{3}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{dd^{2}}{c^{3}}c^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{d^{3}}{c^{3}}c^{3})$$

Cancel out c^{3} and c^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}d}(d^{3})$$

The derivative of ax^{n} is nax^{n-1}.

$$3d^{3-1}$$

Subtract 1 from 3.

$$3d^{2}$$