Solve (3a^{2}*(6b^{3}:3b^{2}))*(7b^2*a^3):(42a^5b^3)=

$$( 3 a ^ { 2 } \cdot ( 6 b ^ { 3 } : 3 b ^ { 2 } ) ) \cdot ( 7 b ^ { 2 } \cdot a ^ { 3 } ) : ( 42 a ^ { 5 } b ^ { 3 } ) =$$

Evaluate

$$b^{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{3a^{5}\times \frac{6b^{3}}{3}b^{2}\times 7b^{2}}{42a^{5}b^{3}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{3a^{5}\times \frac{6b^{3}}{3}b^{4}\times 7}{42a^{5}b^{3}}$$

Divide 6b^{3} by 3 to get 2b^{3}.

$$\frac{3a^{5}\times 2b^{3}b^{4}\times 7}{42a^{5}b^{3}}$$

Multiply 3 and 2 to get 6.

$$\frac{6a^{5}b^{3}b^{4}\times 7}{42a^{5}b^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$\frac{6a^{5}b^{7}\times 7}{42a^{5}b^{3}}$$

Multiply 6 and 7 to get 42.

$$\frac{42a^{5}b^{7}}{42a^{5}b^{3}}$$

Cancel out 42b^{3}a^{5} in both numerator and denominator.

$$b^{4}$$

Differentiate w.r.t. b

$$4b^{3}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{3a^{5}\times \frac{6b^{3}}{3}b^{2}\times 7b^{2}}{42a^{5}b^{3}})$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{3a^{5}\times \frac{6b^{3}}{3}b^{4}\times 7}{42a^{5}b^{3}})$$

Divide 6b^{3} by 3 to get 2b^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{3a^{5}\times 2b^{3}b^{4}\times 7}{42a^{5}b^{3}})$$

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{6a^{5}b^{3}b^{4}\times 7}{42a^{5}b^{3}})$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{6a^{5}b^{7}\times 7}{42a^{5}b^{3}})$$

Multiply 6 and 7 to get 42.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{42a^{5}b^{7}}{42a^{5}b^{3}})$$

Cancel out 42b^{3}a^{5} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(b^{4})$$

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

$$4b^{4-1}$$

Subtract 1 from 4.

$$4b^{3}$$