Solve 140C_{100}(0.7)^100(0.3)^40

$$140 C _ { 100 } ( 0.7 ) ^ { 100 } ( 0.3 ) ^ { 40 }$$

Evaluate

$$\frac{275265783374371553065124533054914728242223689931844506628165353151480056199221597284707751906666790921607C_{100}}{5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Solution Steps

Calculate 0.7 to the power of 100 and get 0.0000000000000003234476509624757991344647769100216810857203198904625400933895331391691459636928060001.

$$140C_{100}\times 0.0000000000000003234476509624757991344647769100216810857203198904625400933895331391691459636928060001\times 0.3^{40}$$

Multiply 140 and 0.0000000000000003234476509624757991344647769100216810857203198904625400933895331391691459636928060001 to get 0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014.

$$0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014C_{100}\times 0.3^{40}$$

Calculate 0.3 to the power of 40 and get 0.0000000000000000000012157665459056928801.

$$0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014C_{100}\times 0.0000000000000000000012157665459056928801$$

Multiply 0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014 and 0.0000000000000000000012157665459056928801 to get 0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214.

$$0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214C_{100}$$

Differentiate w.r.t. C_100

$$0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214$$

Steps Using Definition of a Derivative

Calculate 0.7 to the power of 100 and get 0.0000000000000003234476509624757991344647769100216810857203198904625400933895331391691459636928060001.

$$\frac{\mathrm{d}}{\mathrm{d}C_{100}}(140C_{100}\times 0.0000000000000003234476509624757991344647769100216810857203198904625400933895331391691459636928060001\times 0.3^{40})$$

Multiply 140 and 0.0000000000000003234476509624757991344647769100216810857203198904625400933895331391691459636928060001 to get 0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014.

$$\frac{\mathrm{d}}{\mathrm{d}C_{100}}(0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014C_{100}\times 0.3^{40})$$

Calculate 0.3 to the power of 40 and get 0.0000000000000000000012157665459056928801.

$$\frac{\mathrm{d}}{\mathrm{d}C_{100}}(0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014C_{100}\times 0.0000000000000000000012157665459056928801)$$

Multiply 0.000000000000045282671134746611878825068767403035352000844784664755613074534639483680434916992840014 and 0.0000000000000000000012157665459056928801 to get 0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214.

$$\frac{\mathrm{d}}{\mathrm{d}C_{100}}(0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214C_{100})$$

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

$$0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214C_{100}^{1-1}$$

Subtract 1 from 1.

$$0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214C_{100}^{0}$$

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

$$0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214\times 1$$

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

$$0.0000000000000000000000000000000000550531566748743106130249066109829456484447379863689013256330706302960112398443194569415503813333581843214$$