Solve frac{2x^{4}*(x^2y^4)^-1}{2yx^2}

$$\frac { 2 x ^ { 4 } \cdot ( x ^ { 2 } y ^ { 4 } ) ^ { - 1 } } { 2 y x ^ { 2 } }$$

Evaluate

$$\frac{1}{y^{5}}$$

Solution Steps

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

$$\frac{\frac{1}{x^{2}y^{4}}x^{2}}{y}$$

Express \frac{1}{x^{2}y^{4}}x^{2} as a single fraction.

$$\frac{\frac{x^{2}}{x^{2}y^{4}}}{y}$$

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

$$\frac{\frac{1}{y^{4}}}{y}$$

Express \frac{\frac{1}{y^{4}}}{y} as a single fraction.

$$\frac{1}{y^{4}y}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{1}{y^{5}}$$

Differentiate w.r.t. y

$$-\frac{5}{y^{6}}$$

Steps Using Chain Rule

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

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

Express \frac{1}{x^{2}y^{4}}x^{2} as a single fraction.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{y^{4}}}{y})$$

Express \frac{\frac{1}{y^{4}}}{y} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{4}y})$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{5}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(y^{5}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{5})$$

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}.

$$-\left(y^{5}\right)^{-2}\times 5y^{5-1}$$

Simplify.

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