Solve frac{8x^{-3}y^4x^2}{4x^-1y^4}

$$\frac { 8 x ^ { - 3 } y ^ { 4 x ^ { 2 } } } { 4 x ^ { - 1 } y ^ { 4 } }$$

Differentiate w.r.t. x

$$\frac{4\left(4x^{2}\ln(y)-1\right)y^{4\left(x^{2}-1\right)}}{x^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8y^{4x^{2}}}{4y^{4}}x^{-3-\left(-1\right)})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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

Evaluate

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