Solve 2*10/11*8/9*6/7*4/5*frac{2}{3}x

$$2 \times \frac { 10 } { 11 } \times \frac { 8 } { 9 } \times \frac { 6 } { 7 } \times \frac { 4 } { 5 } \times \frac { 2 } { 3 } x$$

Evaluate

$$\frac{512x}{693}$$

Solution Steps

Express 2\times \frac{10}{11} as a single fraction.

$$\frac{2\times 10}{11}\times \frac{8}{9}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x$$

Multiply 2 and 10 to get 20.

$$\frac{20}{11}\times \frac{8}{9}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x$$

Multiply \frac{20}{11} times \frac{8}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{20\times 8}{11\times 9}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x$$

Do the multiplications in the fraction \frac{20\times 8}{11\times 9}.

$$\frac{160}{99}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x$$

Multiply \frac{160}{99} times \frac{6}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{160\times 6}{99\times 7}\times \frac{4}{5}\times \frac{2}{3}x$$

Do the multiplications in the fraction \frac{160\times 6}{99\times 7}.

$$\frac{960}{693}\times \frac{4}{5}\times \frac{2}{3}x$$

Reduce the fraction \frac{960}{693} to lowest terms by extracting and canceling out 3.

$$\frac{320}{231}\times \frac{4}{5}\times \frac{2}{3}x$$

Multiply \frac{320}{231} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{320\times 4}{231\times 5}\times \frac{2}{3}x$$

Do the multiplications in the fraction \frac{320\times 4}{231\times 5}.

$$\frac{1280}{1155}\times \frac{2}{3}x$$

Reduce the fraction \frac{1280}{1155} to lowest terms by extracting and canceling out 5.

$$\frac{256}{231}\times \frac{2}{3}x$$

Multiply \frac{256}{231} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{256\times 2}{231\times 3}x$$

Do the multiplications in the fraction \frac{256\times 2}{231\times 3}.

$$\frac{512}{693}x$$

Differentiate w.r.t. x

$$\frac{512}{693} = 0.7388167388167388$$

Steps Using Definition of a Derivative

Express 2\times \frac{10}{11} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\times 10}{11}\times \frac{8}{9}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x)$$

Multiply 2 and 10 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20}{11}\times \frac{8}{9}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x)$$

Multiply \frac{20}{11} times \frac{8}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20\times 8}{11\times 9}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x)$$

Do the multiplications in the fraction \frac{20\times 8}{11\times 9}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{160}{99}\times \frac{6}{7}\times \frac{4}{5}\times \frac{2}{3}x)$$

Multiply \frac{160}{99} times \frac{6}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{160\times 6}{99\times 7}\times \frac{4}{5}\times \frac{2}{3}x)$$

Do the multiplications in the fraction \frac{160\times 6}{99\times 7}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{960}{693}\times \frac{4}{5}\times \frac{2}{3}x)$$

Reduce the fraction \frac{960}{693} to lowest terms by extracting and canceling out 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{320}{231}\times \frac{4}{5}\times \frac{2}{3}x)$$

Multiply \frac{320}{231} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{320\times 4}{231\times 5}\times \frac{2}{3}x)$$

Do the multiplications in the fraction \frac{320\times 4}{231\times 5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1280}{1155}\times \frac{2}{3}x)$$

Reduce the fraction \frac{1280}{1155} to lowest terms by extracting and canceling out 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{256}{231}\times \frac{2}{3}x)$$

Multiply \frac{256}{231} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{256\times 2}{231\times 3}x)$$

Do the multiplications in the fraction \frac{256\times 2}{231\times 3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{512}{693}x)$$

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

$$\frac{512}{693}x^{1-1}$$

Subtract 1 from 1.

$$\frac{512}{693}x^{0}$$

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

$$\frac{512}{693}\times 1$$

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

$$\frac{512}{693}$$