Solve (0,5y+sqrt{0,36+0,64})*(0,5y-sqrt{0,36+0,64})=

$$( 0,5 y + \sqrt { 0,36 + 0,64 } ) \cdot ( 0,5 y - \sqrt { 0,36 + 0,64 } ) =$$

Evaluate

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

Solution Steps

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(0,5y\right)^{2}-\left(\sqrt{0,36+0,64}\right)^{2}$$

Expand \left(0,5y\right)^{2}.

$$0,5^{2}y^{2}-\left(\sqrt{0,36+0,64}\right)^{2}$$

Calculate 0,5 to the power of 2 and get 0,25.

$$0,25y^{2}-\left(\sqrt{0,36+0,64}\right)^{2}$$

Add 0,36 and 0,64 to get 1.

$$0,25y^{2}-\left(\sqrt{1}\right)^{2}$$

The square of \sqrt{1} is 1.

$$0,25y^{2}-1$$

Differentiate w.r.t. y

$$\frac{y}{2}$$

Steps Using Derivative Rule for Sum

Consider \left(0,5y+\sqrt{0,36+0,64}\right)\left(0,5y-\sqrt{0,36+0,64}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(0,5y\right)^{2}-\left(\sqrt{0,36+0,64}\right)^{2})$$

Expand \left(0,5y\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(0,5^{2}y^{2}-\left(\sqrt{0,36+0,64}\right)^{2})$$

Calculate 0,5 to the power of 2 and get 0,25.

$$\frac{\mathrm{d}}{\mathrm{d}y}(0,25y^{2}-\left(\sqrt{0,36+0,64}\right)^{2})$$

Add 0,36 and 0,64 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(0,25y^{2}-\left(\sqrt{1}\right)^{2})$$

The square of \sqrt{1} is 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(0,25y^{2}-1)$$

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}.

$$2\times 0,25y^{2-1}$$

Multiply 2 times 0,25.

$$0,5y^{2-1}$$

Subtract 1 from 2.

$$0,5y^{1}$$

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

$$0,5y$$