Solve 6.2m/frac{1{2*0.008}+1/2*0.009+1/2*0.01}

$$\frac{ 6.2m }{ \frac{ 1 }{ 2 \cdot 0.008 } + \frac{ 1 }{ 2 \cdot 0.009 } + \frac{ 1 }{ 2 \cdot 0.01 } }$$

Evaluate

$$\frac{558m}{15125}$$

Solution Steps

Multiply 2 and 0.008 to get 0.016.

$$\frac{6.2m}{\frac{1}{0.016}+\frac{1}{2\times 0.009}+\frac{1}{2\times 0.01}}$$

Expand \frac{1}{0.016} by multiplying both numerator and the denominator by 1000.

$$\frac{6.2m}{\frac{1000}{16}+\frac{1}{2\times 0.009}+\frac{1}{2\times 0.01}}$$

Reduce the fraction \frac{1000}{16} to lowest terms by extracting and canceling out 8.

$$\frac{6.2m}{\frac{125}{2}+\frac{1}{2\times 0.009}+\frac{1}{2\times 0.01}}$$

Multiply 2 and 0.009 to get 0.018.

$$\frac{6.2m}{\frac{125}{2}+\frac{1}{0.018}+\frac{1}{2\times 0.01}}$$

Expand \frac{1}{0.018} by multiplying both numerator and the denominator by 1000.

$$\frac{6.2m}{\frac{125}{2}+\frac{1000}{18}+\frac{1}{2\times 0.01}}$$

Reduce the fraction \frac{1000}{18} to lowest terms by extracting and canceling out 2.

$$\frac{6.2m}{\frac{125}{2}+\frac{500}{9}+\frac{1}{2\times 0.01}}$$

Least common multiple of 2 and 9 is 18. Convert \frac{125}{2} and \frac{500}{9} to fractions with denominator 18.

$$\frac{6.2m}{\frac{1125}{18}+\frac{1000}{18}+\frac{1}{2\times 0.01}}$$

Since \frac{1125}{18} and \frac{1000}{18} have the same denominator, add them by adding their numerators.

$$\frac{6.2m}{\frac{1125+1000}{18}+\frac{1}{2\times 0.01}}$$

Add 1125 and 1000 to get 2125.

$$\frac{6.2m}{\frac{2125}{18}+\frac{1}{2\times 0.01}}$$

Multiply 2 and 0.01 to get 0.02.

$$\frac{6.2m}{\frac{2125}{18}+\frac{1}{0.02}}$$

Expand \frac{1}{0.02} by multiplying both numerator and the denominator by 100.

$$\frac{6.2m}{\frac{2125}{18}+\frac{100}{2}}$$

Divide 100 by 2 to get 50.

$$\frac{6.2m}{\frac{2125}{18}+50}$$

Convert 50 to fraction \frac{900}{18}.

$$\frac{6.2m}{\frac{2125}{18}+\frac{900}{18}}$$

Since \frac{2125}{18} and \frac{900}{18} have the same denominator, add them by adding their numerators.

$$\frac{6.2m}{\frac{2125+900}{18}}$$

Add 2125 and 900 to get 3025.

$$\frac{6.2m}{\frac{3025}{18}}$$

Divide 6.2m by \frac{3025}{18} by multiplying 6.2m by the reciprocal of \frac{3025}{18}.

$$\frac{6.2m\times 18}{3025}$$

Multiply 6.2 and 18 to get 111.6.

$$\frac{111.6m}{3025}$$

Divide 111.6m by 3025 to get \frac{558}{15125}m.

$$\frac{558}{15125}m$$

Differentiate w.r.t. m

$$\frac{558}{15125} = 0.03689256198347107$$

Steps Using Definition of a Derivative

Multiply 2 and 0.008 to get 0.016.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{1}{0.016}+\frac{1}{2\times 0.009}+\frac{1}{2\times 0.01}})$$

Expand \frac{1}{0.016} by multiplying both numerator and the denominator by 1000.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{1000}{16}+\frac{1}{2\times 0.009}+\frac{1}{2\times 0.01}})$$

Reduce the fraction \frac{1000}{16} to lowest terms by extracting and canceling out 8.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{125}{2}+\frac{1}{2\times 0.009}+\frac{1}{2\times 0.01}})$$

Multiply 2 and 0.009 to get 0.018.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{125}{2}+\frac{1}{0.018}+\frac{1}{2\times 0.01}})$$

Expand \frac{1}{0.018} by multiplying both numerator and the denominator by 1000.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{125}{2}+\frac{1000}{18}+\frac{1}{2\times 0.01}})$$

Reduce the fraction \frac{1000}{18} to lowest terms by extracting and canceling out 2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{125}{2}+\frac{500}{9}+\frac{1}{2\times 0.01}})$$

Least common multiple of 2 and 9 is 18. Convert \frac{125}{2} and \frac{500}{9} to fractions with denominator 18.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{1125}{18}+\frac{1000}{18}+\frac{1}{2\times 0.01}})$$

Since \frac{1125}{18} and \frac{1000}{18} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{1125+1000}{18}+\frac{1}{2\times 0.01}})$$

Add 1125 and 1000 to get 2125.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{2125}{18}+\frac{1}{2\times 0.01}})$$

Multiply 2 and 0.01 to get 0.02.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{2125}{18}+\frac{1}{0.02}})$$

Expand \frac{1}{0.02} by multiplying both numerator and the denominator by 100.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{2125}{18}+\frac{100}{2}})$$

Divide 100 by 2 to get 50.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{2125}{18}+50})$$

Convert 50 to fraction \frac{900}{18}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{2125}{18}+\frac{900}{18}})$$

Since \frac{2125}{18} and \frac{900}{18} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{2125+900}{18}})$$

Add 2125 and 900 to get 3025.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m}{\frac{3025}{18}})$$

Divide 6.2m by \frac{3025}{18} by multiplying 6.2m by the reciprocal of \frac{3025}{18}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6.2m\times 18}{3025})$$

Multiply 6.2 and 18 to get 111.6.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{111.6m}{3025})$$

Divide 111.6m by 3025 to get \frac{558}{15125}m.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{558}{15125}m)$$

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

$$\frac{558}{15125}m^{1-1}$$

Subtract 1 from 1.

$$\frac{558}{15125}m^{0}$$

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

$$\frac{558}{15125}\times 1$$

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

$$\frac{558}{15125}$$