Solve (5sqrt{5}y-2sqrt{45})div(-sqrt{5})

$$( 5 \sqrt { 5 } y - 2 \sqrt { 45 } ) \div ( - \sqrt { 5 } )$$

Evaluate

$$6-5y$$

Short Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{5\sqrt{5}y-2\times 3\sqrt{5}}{-\sqrt{5}}$$

Multiply -2 and 3 to get -6.

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

Rationalize the denominator of \frac{5\sqrt{5}y-6\sqrt{5}}{-\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(5\sqrt{5}y-6\sqrt{5}\right)\sqrt{5}}{-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5\sqrt{5}y-6\sqrt{5}\right)\sqrt{5}}{-5}$$

Use the distributive property to multiply 5\sqrt{5}y-6\sqrt{5} by \sqrt{5}.

$$\frac{5y\left(\sqrt{5}\right)^{2}-6\left(\sqrt{5}\right)^{2}}{-5}$$

The square of \sqrt{5} is 5.

$$\frac{5y\times 5-6\left(\sqrt{5}\right)^{2}}{-5}$$

Multiply 5 and 5 to get 25.

$$\frac{25y-6\left(\sqrt{5}\right)^{2}}{-5}$$

The square of \sqrt{5} is 5.

$$\frac{25y-6\times 5}{-5}$$

Multiply -6 and 5 to get -30.

$$\frac{25y-30}{-5}$$

Divide each term of 25y-30 by -5 to get 6-5y.

$$6-5y$$

Differentiate w.r.t. y

$$-5$$

Steps Using Derivative Rule for Sum

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5\sqrt{5}y-2\times 3\sqrt{5}}{-\sqrt{5}})$$

Multiply -2 and 3 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5\sqrt{5}y-6\sqrt{5}}{-\sqrt{5}})$$

Rationalize the denominator of \frac{5\sqrt{5}y-6\sqrt{5}}{-\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(5\sqrt{5}y-6\sqrt{5}\right)\sqrt{5}}{-\left(\sqrt{5}\right)^{2}})$$

The square of \sqrt{5} is 5.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(5\sqrt{5}y-6\sqrt{5}\right)\sqrt{5}}{-5})$$

Use the distributive property to multiply 5\sqrt{5}y-6\sqrt{5} by \sqrt{5}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5y\left(\sqrt{5}\right)^{2}-6\left(\sqrt{5}\right)^{2}}{-5})$$

The square of \sqrt{5} is 5.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5y\times 5-6\left(\sqrt{5}\right)^{2}}{-5})$$

Multiply 5 and 5 to get 25.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{25y-6\left(\sqrt{5}\right)^{2}}{-5})$$

The square of \sqrt{5} is 5.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{25y-6\times 5}{-5})$$

Multiply -6 and 5 to get -30.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{25y-30}{-5})$$

Divide each term of 25y-30 by -5 to get 6-5y.

$$\frac{\mathrm{d}}{\mathrm{d}y}(6-5y)$$

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}.

$$-5y^{1-1}$$

Subtract 1 from 1.

$$-5y^{0}$$

For any term t except 0, t^{0}=1.

$$-5$$