Solve B/[4/7+5B/252]=

$$B / [ 4 / 7 + 5 B / 252 ] =$$

Evaluate

$$\frac{252B}{5B+144}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 252 is 252. Multiply \frac{4}{7} times \frac{36}{36}.

$$\frac{B}{\frac{4\times 36}{252}+\frac{5B}{252}}$$

Since \frac{4\times 36}{252} and \frac{5B}{252} have the same denominator, add them by adding their numerators.

$$\frac{B}{\frac{4\times 36+5B}{252}}$$

Do the multiplications in 4\times 36+5B.

$$\frac{B}{\frac{144+5B}{252}}$$

Divide B by \frac{144+5B}{252} by multiplying B by the reciprocal of \frac{144+5B}{252}.

$$\frac{B\times 252}{144+5B}$$

Differentiate w.r.t. B

$$\frac{36288}{\left(5B+144\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 252 is 252. Multiply \frac{4}{7} times \frac{36}{36}.

$$\frac{\mathrm{d}}{\mathrm{d}B}(\frac{B}{\frac{4\times 36}{252}+\frac{5B}{252}})$$

Since \frac{4\times 36}{252} and \frac{5B}{252} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}B}(\frac{B}{\frac{4\times 36+5B}{252}})$$

Do the multiplications in 4\times 36+5B.

$$\frac{\mathrm{d}}{\mathrm{d}B}(\frac{B}{\frac{144+5B}{252}})$$

Divide B by \frac{144+5B}{252} by multiplying B by the reciprocal of \frac{144+5B}{252}.

$$\frac{\mathrm{d}}{\mathrm{d}B}(\frac{B\times 252}{144+5B})$$

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.

$$\frac{\left(5B^{1}+144\right)\frac{\mathrm{d}}{\mathrm{d}B}(252B^{1})-252B^{1}\frac{\mathrm{d}}{\mathrm{d}B}(5B^{1}+144)}{\left(5B^{1}+144\right)^{2}}$$

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}.

$$\frac{\left(5B^{1}+144\right)\times 252B^{1-1}-252B^{1}\times 5B^{1-1}}{\left(5B^{1}+144\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(5B^{1}+144\right)\times 252B^{0}-252B^{1}\times 5B^{0}}{\left(5B^{1}+144\right)^{2}}$$

Expand using distributive property.

$$\frac{5B^{1}\times 252B^{0}+144\times 252B^{0}-252B^{1}\times 5B^{0}}{\left(5B^{1}+144\right)^{2}}$$

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

$$\frac{5\times 252B^{1}+144\times 252B^{0}-252\times 5B^{1}}{\left(5B^{1}+144\right)^{2}}$$

Do the arithmetic.

$$\frac{1260B^{1}+36288B^{0}-1260B^{1}}{\left(5B^{1}+144\right)^{2}}$$

Combine like terms.

$$\frac{\left(1260-1260\right)B^{1}+36288B^{0}}{\left(5B^{1}+144\right)^{2}}$$

Subtract 1260 from 1260.

$$\frac{36288B^{0}}{\left(5B^{1}+144\right)^{2}}$$

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

$$\frac{36288B^{0}}{\left(5B+144\right)^{2}}$$

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

$$\frac{36288\times 1}{\left(5B+144\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{36288}{\left(5B+144\right)^{2}}$$