Solve b/frac{b^2{1-b}}

$$\frac { b } { \frac { b ^ { 2 } } { 1 - b } }$$

Evaluate

$$-1+\frac{1}{b}$$

Solution Steps

Divide b by \frac{b^{2}}{1-b} by multiplying b by the reciprocal of \frac{b^{2}}{1-b}.

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

Cancel out b in both numerator and denominator.

$$\frac{-b+1}{b}$$

Differentiate w.r.t. b

$$-\frac{1}{b^{2}}$$

Steps Using Derivative Rule for Product

Divide b by \frac{b^{2}}{1-b} by multiplying b by the reciprocal of \frac{b^{2}}{1-b}.

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

Cancel out b in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{-b+1}{b})$$

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

$$\left(-b^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{b})+\frac{1}{b}\frac{\mathrm{d}}{\mathrm{d}b}(-b^{1}+1)$$

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}.

$$\left(-b^{1}+1\right)\left(-1\right)b^{-1-1}+\frac{1}{b}\left(-1\right)b^{1-1}$$

Simplify.

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

Multiply -b^{1}+1 times -b^{-2}.

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

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

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

Simplify.

$$\frac{1}{b}-b^{-2}-\frac{1}{b}$$

Steps Using Derivative Rule for Quotient

Divide b by \frac{b^{2}}{1-b} by multiplying b by the reciprocal of \frac{b^{2}}{1-b}.

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

Cancel out b in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{-b+1}{b})$$

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.

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

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}.

$$\frac{b^{1}\left(-1\right)b^{1-1}-\left(-b^{1}+1\right)b^{1-1}}{\left(b^{1}\right)^{2}}$$

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Remove unnecessary parentheses.

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

Combine like terms.

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

Subtract -1 from -1.

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

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$-\frac{b^{0}}{1^{2}b^{2}}$$

Raise 1 to the power 2.

$$-\frac{b^{0}}{b^{2}}$$

Multiply 1 times 2.

$$\frac{-b^{0}}{b^{2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Do the arithmetic.

$$-b^{-2}$$