Solve 5x^{4}y^2*-2x^-1y^2/y^4

$$5 x ^ { 4 } y ^ { 2 } \times \frac { - 2 x ^ { - 1 } y ^ { 2 } } { y ^ { 4 } }$$

Evaluate

$$-10x^{3}$$

Solution Steps

Cancel out y^{2} in both numerator and denominator.

$$5x^{4}y^{2}\times \frac{-2\times \frac{1}{x}}{y^{2}}$$

Express -2\times \frac{1}{x} as a single fraction.

$$5x^{4}y^{2}\times \frac{\frac{-2}{x}}{y^{2}}$$

Express \frac{\frac{-2}{x}}{y^{2}} as a single fraction.

$$5x^{4}y^{2}\times \frac{-2}{xy^{2}}$$

Express 5\times \frac{-2}{xy^{2}} as a single fraction.

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

Multiply 5 and -2 to get -10.

$$\frac{-10}{xy^{2}}x^{4}y^{2}$$

Express \frac{-10}{xy^{2}}x^{4} as a single fraction.

$$\frac{-10x^{4}}{xy^{2}}y^{2}$$

Cancel out x in both numerator and denominator.

$$\frac{-10x^{3}}{y^{2}}y^{2}$$

Cancel out y^{2} and y^{2}.

$$-10x^{3}$$

Differentiate w.r.t. x

$$-30x^{2}$$

Steps Using Definition of a Derivative

Cancel out y^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{4}y^{2}\times \frac{-2\times \frac{1}{x}}{y^{2}})$$

Express -2\times \frac{1}{x} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{4}y^{2}\times \frac{\frac{-2}{x}}{y^{2}})$$

Express \frac{\frac{-2}{x}}{y^{2}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{4}y^{2}\times \frac{-2}{xy^{2}})$$

Express 5\times \frac{-2}{xy^{2}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(-2\right)}{xy^{2}}x^{4}y^{2})$$

Multiply 5 and -2 to get -10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-10}{xy^{2}}x^{4}y^{2})$$

Express \frac{-10}{xy^{2}}x^{4} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-10x^{4}}{xy^{2}}y^{2})$$

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-10x^{3}}{y^{2}}y^{2})$$

Cancel out y^{2} and y^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-10x^{3})$$

The derivative of ax^{n} is nax^{n-1}.

$$3\left(-10\right)x^{3-1}$$

Multiply 3 times -10.

$$-30x^{3-1}$$

Subtract 1 from 3.

$$-30x^{2}$$