Solve 49b^2/49b^2-4div49b/63b+18

$$\frac { 49 b ^ { 2 } } { 49 b ^ { 2 } - 4 } \div \frac { 49 b } { 63 b + 18 }$$

Evaluate

$$\frac{9b}{7b-2}$$

Short Solution Steps

Divide \frac{49b^{2}}{49b^{2}-4} by \frac{49b}{63b+18} by multiplying \frac{49b^{2}}{49b^{2}-4} by the reciprocal of \frac{49b}{63b+18}.

$$\frac{49b^{2}\left(63b+18\right)}{\left(49b^{2}-4\right)\times 49b}$$

Cancel out 49b in both numerator and denominator.

$$\frac{b\left(63b+18\right)}{49b^{2}-4}$$

Factor the expressions that are not already factored.

$$\frac{9b\left(7b+2\right)}{\left(7b-2\right)\left(7b+2\right)}$$

Cancel out 7b+2 in both numerator and denominator.

$$\frac{9b}{7b-2}$$

Differentiate w.r.t. b

$$-\frac{18}{\left(7b-2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Divide \frac{49b^{2}}{49b^{2}-4} by \frac{49b}{63b+18} by multiplying \frac{49b^{2}}{49b^{2}-4} by the reciprocal of \frac{49b}{63b+18}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{49b^{2}\left(63b+18\right)}{\left(49b^{2}-4\right)\times 49b})$$

Cancel out 49b in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b\left(63b+18\right)}{49b^{2}-4})$$

Factor the expressions that are not already factored in \frac{b\left(63b+18\right)}{49b^{2}-4}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{9b\left(7b+2\right)}{\left(7b-2\right)\left(7b+2\right)})$$

Cancel out 7b+2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{9b}{7b-2})$$

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.

$$\frac{\left(7b^{1}-2\right)\frac{\mathrm{d}}{\mathrm{d}b}(9b^{1})-9b^{1}\frac{\mathrm{d}}{\mathrm{d}b}(7b^{1}-2)}{\left(7b^{1}-2\right)^{2}}$$

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}.

$$\frac{\left(7b^{1}-2\right)\times 9b^{1-1}-9b^{1}\times 7b^{1-1}}{\left(7b^{1}-2\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(7b^{1}-2\right)\times 9b^{0}-9b^{1}\times 7b^{0}}{\left(7b^{1}-2\right)^{2}}$$

Expand using distributive property.

$$\frac{7b^{1}\times 9b^{0}-2\times 9b^{0}-9b^{1}\times 7b^{0}}{\left(7b^{1}-2\right)^{2}}$$

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

$$\frac{7\times 9b^{1}-2\times 9b^{0}-9\times 7b^{1}}{\left(7b^{1}-2\right)^{2}}$$

Do the arithmetic.

$$\frac{63b^{1}-18b^{0}-63b^{1}}{\left(7b^{1}-2\right)^{2}}$$

Combine like terms.

$$\frac{\left(63-63\right)b^{1}-18b^{0}}{\left(7b^{1}-2\right)^{2}}$$

Subtract 63 from 63.

$$\frac{-18b^{0}}{\left(7b^{1}-2\right)^{2}}$$

For any term t, t^{1}=t.

$$\frac{-18b^{0}}{\left(7b-2\right)^{2}}$$

For any term t except 0, t^{0}=1.

$$\frac{-18}{\left(7b-2\right)^{2}}$$