Solve frac{3^{5}*10^{5}*25*9^2b^3}{5*6^5*9^6}

$$\frac { 3 ^ { 5 } \times 10 ^ { 5 } \times 25 \times 9 ^ { 2 } b ^ { 3 } } { 5 \times 6 ^ { 5 } \times 9 ^ { 6 } }$$

Evaluate

$$\frac{15625b^{3}}{6561}$$

Solution Steps

Cancel out 5\times 9^{2} in both numerator and denominator.

$$\frac{5\times 3^{5}\times 10^{5}b^{3}}{6^{5}\times 9^{4}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{5\times 243\times 10^{5}b^{3}}{6^{5}\times 9^{4}}$$

Multiply 5 and 243 to get 1215.

$$\frac{1215\times 10^{5}b^{3}}{6^{5}\times 9^{4}}$$

Calculate 10 to the power of 5 and get 100000.

$$\frac{1215\times 100000b^{3}}{6^{5}\times 9^{4}}$$

Multiply 1215 and 100000 to get 121500000.

$$\frac{121500000b^{3}}{6^{5}\times 9^{4}}$$

Calculate 6 to the power of 5 and get 7776.

$$\frac{121500000b^{3}}{7776\times 9^{4}}$$

Calculate 9 to the power of 4 and get 6561.

$$\frac{121500000b^{3}}{7776\times 6561}$$

Multiply 7776 and 6561 to get 51018336.

$$\frac{121500000b^{3}}{51018336}$$

Divide 121500000b^{3} by 51018336 to get \frac{15625}{6561}b^{3}.

$$\frac{15625}{6561}b^{3}$$

Differentiate w.r.t. b

$$\frac{15625b^{2}}{2187}$$

Steps Using Definition of a Derivative

Cancel out 5\times 9^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{5\times 3^{5}\times 10^{5}b^{3}}{6^{5}\times 9^{4}})$$

Calculate 3 to the power of 5 and get 243.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{5\times 243\times 10^{5}b^{3}}{6^{5}\times 9^{4}})$$

Multiply 5 and 243 to get 1215.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1215\times 10^{5}b^{3}}{6^{5}\times 9^{4}})$$

Calculate 10 to the power of 5 and get 100000.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1215\times 100000b^{3}}{6^{5}\times 9^{4}})$$

Multiply 1215 and 100000 to get 121500000.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{121500000b^{3}}{6^{5}\times 9^{4}})$$

Calculate 6 to the power of 5 and get 7776.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{121500000b^{3}}{7776\times 9^{4}})$$

Calculate 9 to the power of 4 and get 6561.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{121500000b^{3}}{7776\times 6561})$$

Multiply 7776 and 6561 to get 51018336.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{121500000b^{3}}{51018336})$$

Divide 121500000b^{3} by 51018336 to get \frac{15625}{6561}b^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{15625}{6561}b^{3})$$

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

$$3\times \frac{15625}{6561}b^{3-1}$$

Multiply 3 times \frac{15625}{6561}.

$$\frac{15625}{2187}b^{3-1}$$

Subtract 1 from 3.

$$\frac{15625}{2187}b^{2}$$