What is the derivative of 10^x?

Pam Stokes 2022-01-02 Answered
What is the derivative of \(\displaystyle{10}^{{x}}?\)

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Expert Answer

Jeremy Merritt
Answered 2022-01-03 Author has 2507 answers
There is a rule for differentiating these functions
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{a}^{{u}}\right]}={\left({\ln{{a}}}\right)}\cdot{\left({a}^{{u}}\right)}\cdot{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\)
Notice that for our problem a=10 and u=x so let's plug in what we know.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{10}^{{x}}\right]}={\left({\ln{{10}}}\right)}\cdot{\left({10}^{{x}}\right)}\cdot{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\)
\(\displaystyle\text{if }\ {u}={x}\ \text{ then, }\ {\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={1}\)
because of the power rule: \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{x}^{{n}}\right]}={n}\cdot{x}^{{{n}-{1}}}\)
so, back to our problem, \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{10}^{{x}}\right]}={\left({\ln{{10}}}\right)}\cdot{\left({10}^{{x}}\right)}\cdot{\left({1}\right)}\)
which simplifies to \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{10}^{{x}}\right]}={\left({\ln{{10}}}\right)}\cdot{\left({10}^{{x}}\right)}\)
This would work the same if u was something more complicated than x. A lot of calculus deals with the ability to relate the given problem to one of the rules of differentiation. Often we have to alter the way the problem looks before we can begin, however that was not the case with this problem.
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Stella Calderon
Answered 2022-01-04 Author has 1124 answers
Taking natural log on both sides,
\(\displaystyle{\ln{{y}}}={{\ln{{10}}}^{{x}}}\)
We have, \(\displaystyle{\ln{{\left({a}^{{b}}\right)}}}={b}{\ln{{a}}}\)
Likewise, \(\displaystyle{\ln{{\left({10}^{{x}}\right)}}}={x}{\ln{{10}}}\)
So, \(\displaystyle{\ln{{y}}}={x}{\ln{{10}}}\)
Now differentiate with respect to \(\displaystyle{x}\)
\(\displaystyle{\frac{{{1}}}{{{y}}}}\cdot{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\ln{{10}}}\)
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={y}{\ln{{10}}}\)
Plug in \(\displaystyle{y}={10}^{{x}}\)
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={10}^{{x}}{\ln{{10}}}\)
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Vasquez
Answered 2022-01-09 Author has 8850 answers

Using, \(\frac{d}{dx}(a^x)=a^x\log a\)
\(\frac{d}{dx}(10^x)=10^x\log10\)
This is the derivative of \(y=10^x\)

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