Solve frac{5^{-7}x^{-1}y^3}{5^-4x^-4y^6}

$$\frac { 5 ^ { - 7 } x ^ { - 1 } y ^ { 3 } } { 5 ^ { - 4 } x ^ { - 4 } y ^ { 6 } }$$

Evaluate

$$\frac{\left(\frac{x}{y}\right)^{3}}{125}$$

Solution Steps

Cancel out y^{3} in both numerator and denominator.

$$\frac{5^{-7}\times \frac{1}{x}}{5^{-4}x^{-4}y^{3}}$$

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

$$\frac{5^{-7}x^{3}}{5^{-4}y^{3}}$$

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

$$\frac{x^{3}}{5^{3}y^{3}}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{x^{3}}{125y^{3}}$$

Differentiate w.r.t. x

$$\frac{3x^{2}}{125y^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{3}}{78125\times \frac{y^{6}}{625}}x^{-1-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{125y^{3}}x^{3})$$

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}.

$$3\times \frac{1}{125y^{3}}x^{3-1}$$

Do the arithmetic.

$$\frac{3}{125y^{3}}x^{2}$$