Solve 2m^2*7m*4/3m^2*14m+8

$$\frac { 2 m ^ { 2 } \cdot 7 m \cdot 4 } { 3 m ^ { 2 } \cdot 14 m + 8 }$$

Evaluate

$$\frac{28m^{3}}{21m^{3}+4}$$

Short Solution Steps

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

$$\frac{2m^{3}\times 7\times 4}{3m^{2}\times 14m+8}$$

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

$$\frac{2m^{3}\times 7\times 4}{3m^{3}\times 14+8}$$

Multiply 2 and 7 to get 14.

$$\frac{14m^{3}\times 4}{3m^{3}\times 14+8}$$

Multiply 14 and 4 to get 56.

$$\frac{56m^{3}}{3m^{3}\times 14+8}$$

Multiply 3 and 14 to get 42.

$$\frac{56m^{3}}{42m^{3}+8}$$

Factor the expressions that are not already factored.

$$\frac{56m^{3}}{2\left(21m^{3}+4\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{28m^{3}}{21m^{3}+4}$$

Differentiate w.r.t. m

$$336\times \left(\frac{m}{21m^{3}+4}\right)^{2}$$

Steps Using Derivative Rule for Quotient

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{2m^{3}\times 7\times 4}{3m^{2}\times 14m+8})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{2m^{3}\times 7\times 4}{3m^{3}\times 14+8})$$

Multiply 2 and 7 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{14m^{3}\times 4}{3m^{3}\times 14+8})$$

Multiply 14 and 4 to get 56.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{56m^{3}}{3m^{3}\times 14+8})$$

Multiply 3 and 14 to get 42.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{56m^{3}}{42m^{3}+8})$$

Factor the expressions that are not already factored in \frac{56m^{3}}{42m^{3}+8}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{56m^{3}}{2\left(21m^{3}+4\right)})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{28m^{3}}{21m^{3}+4})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(21m^{3}+4\right)\frac{\mathrm{d}}{\mathrm{d}m}(28m^{3})-28m^{3}\frac{\mathrm{d}}{\mathrm{d}m}(21m^{3}+4)}{\left(21m^{3}+4\right)^{2}}$$

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}.

$$\frac{\left(21m^{3}+4\right)\times 3\times 28m^{3-1}-28m^{3}\times 3\times 21m^{3-1}}{\left(21m^{3}+4\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(21m^{3}+4\right)\times 84m^{2}-28m^{3}\times 63m^{2}}{\left(21m^{3}+4\right)^{2}}$$

Expand using distributive property.

$$\frac{21m^{3}\times 84m^{2}+4\times 84m^{2}-28m^{3}\times 63m^{2}}{\left(21m^{3}+4\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{21\times 84m^{3+2}+4\times 84m^{2}-28\times 63m^{3+2}}{\left(21m^{3}+4\right)^{2}}$$

Do the arithmetic.

$$\frac{1764m^{5}+336m^{2}-1764m^{5}}{\left(21m^{3}+4\right)^{2}}$$

Combine like terms.

$$\frac{\left(1764-1764\right)m^{5}+336m^{2}}{\left(21m^{3}+4\right)^{2}}$$

Subtract 1764 from 1764.

$$\frac{336m^{2}}{\left(21m^{3}+4\right)^{2}}$$