Solve 9b/45b^11

$$\frac { 9 b } { 45 b ^ { 11 } }$$

Evaluate

$$\frac{1}{5b^{10}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(9b^{1}\right)^{1}\times \frac{1}{45b^{11}}$$

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

$$9^{1}\left(b^{1}\right)^{1}\times \frac{1}{45}\times \frac{1}{b^{11}}$$

Use the Commutative Property of Multiplication.

$$9^{1}\times \frac{1}{45}\left(b^{1}\right)^{1}\times \frac{1}{b^{11}}$$

To raise a power to another power, multiply the exponents.

$$9^{1}\times \frac{1}{45}b^{1}b^{11\left(-1\right)}$$

Multiply 11 times -1.

$$9^{1}\times \frac{1}{45}b^{1}b^{-11}$$

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

$$9^{1}\times \frac{1}{45}b^{1-11}$$

Add the exponents 1 and -11.

$$9^{1}\times \frac{1}{45}b^{-10}$$

Raise 9 to the power 1.

$$9\times \frac{1}{45}b^{-10}$$

Multiply 9 times \frac{1}{45}.

$$\frac{1}{5}b^{-10}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{9^{1}b^{1}}{45^{1}b^{11}}$$

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

$$\frac{9^{1}b^{1-11}}{45^{1}}$$

Subtract 11 from 1.

$$\frac{9^{1}b^{-10}}{45^{1}}$$

Reduce the fraction \frac{9}{45} to lowest terms by extracting and canceling out 9.

$$\frac{1}{5}b^{-10}$$

Differentiate w.r.t. b

$$-\frac{2}{b^{11}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{9}{45}b^{1-11})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{5}b^{-10})$$

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}.

$$-10\times \frac{1}{5}b^{-10-1}$$

Do the arithmetic.

$$-2b^{-11}$$