Solve 30x^6y^2/45x^4y^2

$$\frac { 30 x ^ { 6 } y ^ { 2 } } { 45 x ^ { 4 } y ^ { 2 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}x^{6}y^{2}}{45^{1}x^{4}y^{2}}$$

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

$$\frac{30^{1}}{45^{1}}x^{6-4}y^{2-2}$$

Subtract 4 from 6.

$$\frac{30^{1}}{45^{1}}x^{2}y^{2-2}$$

Subtract 2 from 2.

$$\frac{30^{1}}{45^{1}}x^{2}y^{0}$$

For any number a except 0, a^{0}=1.

$$\frac{30^{1}}{45^{1}}x^{2}$$

Reduce the fraction \frac{30}{45} to lowest terms by extracting and canceling out 15.

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

Differentiate w.r.t. x

$$\frac{4x}{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{30y^{2}}{45y^{2}}x^{6-4})$$

Do the arithmetic.

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

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}.

$$2\times \frac{2}{3}x^{2-1}$$

Do the arithmetic.

$$\frac{4}{3}x^{1}$$

For any term t, t^{1}=t.

$$\frac{4}{3}x$$