Solve T-10(1/2)+6*(1/sqrt{2})^2

$$T - 10 ( \frac { 1 } { 2 } ) + 6 \times ( \frac { 1 } { \sqrt { 2 } } ) ^ { 2 }$$

Evaluate

$$T-2$$

Solution Steps

Multiply 10 and \frac{1}{2} to get 5.

$$T-5+6\times \left(\frac{1}{\sqrt{2}}\right)^{2}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Express 6\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply T-5 times \frac{2^{2}}{2^{2}}.

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

Since \frac{\left(T-5\right)\times 2^{2}}{2^{2}} and \frac{6\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{2^{2}}{2^{2}}.

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

Since -\frac{5\times 2^{2}}{2^{2}} and \frac{6\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

The square of \sqrt{2} is 2.

$$T-5+\frac{6\times 2}{2^{2}}$$

Multiply 6 and 2 to get 12.

$$T-5+\frac{12}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$T-5+\frac{12}{4}$$

Divide 12 by 4 to get 3.

$$T-5+3$$

Add -5 and 3 to get -2.

$$T-2$$

Differentiate w.r.t. T

$$1$$

Steps Using Derivative Rule for Sum

Multiply 10 and \frac{1}{2} to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T-5+6\times \left(\frac{1}{\sqrt{2}}\right)^{2})$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Express 6\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply T-5 times \frac{2^{2}}{2^{2}}.

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

Since \frac{\left(T-5\right)\times 2^{2}}{2^{2}} and \frac{6\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{2^{2}}{2^{2}}.

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

Since -\frac{5\times 2^{2}}{2^{2}} and \frac{6\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T+\frac{-5\times 4+6\left(\sqrt{2}\right)^{2}}{2^{2}})$$

Multiply -5 and 4 to get -20.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T+\frac{-20+6\left(\sqrt{2}\right)^{2}}{2^{2}})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T+\frac{-20+6\times 2}{2^{2}})$$

Multiply 6 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T+\frac{-20+12}{2^{2}})$$

Add -20 and 12 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T+\frac{-8}{2^{2}})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T+\frac{-8}{4})$$

Divide -8 by 4 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}T}(T-2)$$

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}.

$$T^{1-1}$$

Subtract 1 from 1.

$$T^{0}$$

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

$$1$$