Solve frac{(32x^{2}b^{8})(-5x^{-4}b)}{(-4x^3b)(-2x^2b^2)(5x^3b)}

$$\frac { ( 32 x ^ { 2 } b ^ { 8 } ) ( - 5 x ^ { - 4 } b ) } { ( - 4 x ^ { 3 } b ) ( - 2 x ^ { 2 } b ^ { 2 } ) ( 5 x ^ { 3 } b ) }$$

Evaluate

$$-\frac{4b^{5}}{x^{10}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and -4 to get -2.

$$\frac{32x^{-2}b^{8}\left(-5\right)b}{-4x^{3}b\left(-2\right)x^{2}b^{2}\times 5x^{3}b}$$

To multiply powers of the same base, add their exponents. Add 8 and 1 to get 9.

$$\frac{32x^{-2}b^{9}\left(-5\right)}{-4x^{3}b\left(-2\right)x^{2}b^{2}\times 5x^{3}b}$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{32x^{-2}b^{9}\left(-5\right)}{-4x^{5}b\left(-2\right)b^{2}\times 5x^{3}b}$$

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$\frac{32x^{-2}b^{9}\left(-5\right)}{-4x^{8}b\left(-2\right)b^{2}\times 5b}$$

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

$$\frac{32x^{-2}b^{9}\left(-5\right)}{-4x^{8}b^{3}\left(-2\right)\times 5b}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{32x^{-2}b^{9}\left(-5\right)}{-4x^{8}b^{4}\left(-2\right)\times 5}$$

Cancel out 2\times 4\times 5b^{4} in both numerator and denominator.

$$\frac{-4x^{-2}b^{5}}{-\left(-1\right)x^{8}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{4x^{-2}b^{5}}{-x^{8}}$$

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

$$\frac{4b^{5}}{-x^{10}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{-4b^{5}}{x^{10}}$$

Differentiate w.r.t. x

$$\frac{40b^{5}}{x^{11}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{\frac{160b^{9}}{x^{4}}}{40b^{4}x^{5}}\right)x^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{4b^{5}}{x^{9}}\right)\times \frac{1}{x})$$

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}.

$$-\left(-\frac{4b^{5}}{x^{9}}\right)x^{-1-1}$$

Do the arithmetic.

$$\frac{4b^{5}}{x^{9}}x^{-2}$$