Solve frac{3x^{3}z^{5}*6^8}{2^3x^3*6^4}

$$\frac { 3 x ^ { 3 } z ^ { 5 } \times 6 ^ { 8 } } { 2 ^ { 3 } x ^ { 3 } \times 6 ^ { 4 } }$$

Evaluate

$$486z^{5}$$

Solution Steps

Cancel out 6^{4}x^{3} in both numerator and denominator.

$$\frac{3\times 6^{4}z^{5}}{2^{3}}$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{3\times 1296z^{5}}{2^{3}}$$

Multiply 3 and 1296 to get 3888.

$$\frac{3888z^{5}}{2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{3888z^{5}}{8}$$

Divide 3888z^{5} by 8 to get 486z^{5}.

$$486z^{5}$$

Differentiate w.r.t. z

$$2430z^{4}$$

Steps Using Definition of a Derivative

Cancel out 6^{4}x^{3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3\times 6^{4}z^{5}}{2^{3}})$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3\times 1296z^{5}}{2^{3}})$$

Multiply 3 and 1296 to get 3888.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3888z^{5}}{2^{3}})$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3888z^{5}}{8})$$

Divide 3888z^{5} by 8 to get 486z^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(486z^{5})$$

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

$$5\times 486z^{5-1}$$

Multiply 5 times 486.

$$2430z^{5-1}$$

Subtract 1 from 5.

$$2430z^{4}$$