Solve 24/2quadtext{b}frac{9sqrt{80}}{3sqrt{5}}

$$\frac { 24 } { 2 } \quad \text { b } \frac { 9 \sqrt { 80 } } { 3 \sqrt { 5 } }$$

Evaluate

$$144b$$

Solution Steps

Divide 24 by 2 to get 12.

$$12b\times \frac{9\sqrt{80}}{3\sqrt{5}}$$

Cancel out 3 in both numerator and denominator.

$$12b\times \frac{3\sqrt{80}}{\sqrt{5}}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$12b\times \frac{3\times 4\sqrt{5}}{\sqrt{5}}$$

Multiply 3 and 4 to get 12.

$$12b\times \frac{12\sqrt{5}}{\sqrt{5}}$$

Rationalize the denominator of \frac{12\sqrt{5}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$12b\times \frac{12\sqrt{5}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$12b\times \frac{12\sqrt{5}\sqrt{5}}{5}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$12b\times \frac{12\times 5}{5}$$

Multiply 12 and 5 to get 60.

$$12b\times \frac{60}{5}$$

Divide 60 by 5 to get 12.

$$12b\times 12$$

Multiply 12 and 12 to get 144.

$$144b$$

Differentiate w.r.t. b

$$144$$

Steps Using Definition of a Derivative

Divide 24 by 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{9\sqrt{80}}{3\sqrt{5}})$$

Cancel out 3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{3\sqrt{80}}{\sqrt{5}})$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{3\times 4\sqrt{5}}{\sqrt{5}})$$

Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{12\sqrt{5}}{\sqrt{5}})$$

Rationalize the denominator of \frac{12\sqrt{5}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{12\sqrt{5}\sqrt{5}}{\left(\sqrt{5}\right)^{2}})$$

The square of \sqrt{5} is 5.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{12\sqrt{5}\sqrt{5}}{5})$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{12\times 5}{5})$$

Multiply 12 and 5 to get 60.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times \frac{60}{5})$$

Divide 60 by 5 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}b}(12b\times 12)$$

Multiply 12 and 12 to get 144.

$$\frac{\mathrm{d}}{\mathrm{d}b}(144b)$$

The derivative of ax^{n} is nax^{n-1}.

$$144b^{1-1}$$

Subtract 1 from 1.

$$144b^{0}$$

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

$$144\times 1$$

For any term t, t\times 1=t and 1t=t.

$$144$$