Solve (16x^5left(y^-2right)^-1div(2x{left(y^3right)}^-4

$$(16 { x }^{ 5 } { \left( { y }^{ -2 } \right) }^{ -1 } \div (2x { \left( { y }^{ 3 } \right) }^{ -4 }$$

Evaluate

$$8x^{4}y^{14}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$\frac{16x^{5}y^{2}}{2x\left(y^{3}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\frac{16x^{5}y^{2}}{2xy^{-12}}$$

Cancel out 2x in both numerator and denominator.

$$\frac{8y^{2}x^{4}}{y^{-12}}$$

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

$$8x^{4}y^{14}$$

Differentiate w.r.t. x

$$32x^{3}y^{14}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16y^{2}}{\frac{2}{y^{12}}}x^{5-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8y^{14}x^{4})$$

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}.

$$4\times 8y^{14}x^{4-1}$$

Do the arithmetic.

$$32y^{14}x^{3}$$