Q. \[ \frac{d}{dx}\left(10^x\right) \]
Answer
To find the derivative of \(10^x\), use the rule \( \dfrac{d}{dx}\left(a^x\right)=a^x\ln(a)\) for \(a>0\).
Let \(a=10\). Then
\[
\dfrac{d}{dx}\left(10^x\right)=10^x\ln(10).
\]
Final result: \(10^x\ln(10)\).
Detailed Explanation
We want to find the derivative of the function
\[
f(x)=10^{x}.
\]
Step 1: Use the general exponential rule.
For any positive base \(a>0\) with \(a\neq 1\), the derivative is
\[
\frac{d}{dx}\left(a^{x}\right)=a^{x}\ln(a).
\]
Here, the base is \(a=10\), so we substitute \(a=10\) into the rule.
Step 2: Substitute \(a=10\).
\[
\frac{d}{dx}\left(10^{x}\right)=10^{x}\ln(10).
\]
Final answer:
\[
\frac{d}{dx}\left(10^{x}\right)=10^{x}\ln(10).
\]
See full solution
Calculus FAQ
What is the derivative of \(10^x\) ?
\(\dfrac{d}{dx}\left(10^x\right)=10^x\ln(10)\).
How do you differentiate \(a^x\) for \(a>0\), \(a\neq 1\) ?
\(\dfrac{d}{dx}(a^x)=a^x\ln(a)\).
Can you derive the rule using logarithms or change of base?
Write \(10^x=e^{x\ln(10)}\). Then \(\dfrac{d}{dx}e^{x\ln(10)}=e^{x\ln(10)}\ln(10)=10^x\ln(10)\).
What is \(\dfrac{d}{dx}\left(10^{3x}\right)\) ?
\(\dfrac{d}{dx}\left(10^{3x}\right)=10^{3x}\ln(10)\cdot 3=3\ln(10)\,10^{3x}\).
What is \(\dfrac{d}{dx}\left(5\cdot 10^x\right)\) ?
Use the constant multiple rule: \(\dfrac{d}{dx}(5\cdot 10^x)=5\cdot 10^x\ln(10)=5\ln(10)\,10^x\).
How do you find the derivative of \(\dfrac{1}{10^x}\) ?
\(\dfrac{1}{10^x}=10^{-x}\). So \(\dfrac{d}{dx}(10^{-x})=10^{-x}\ln(10)\cdot (-1)=-\ln(10)\,10^{-x}\).
Try math AI tools for 10^x.
Get step-by-step help now.
Get step-by-step help now.
298,376+ active customers
Math, Geometry, Trigonometry, etc.
Math, Geometry, Trigonometry, etc.
Unlimited AI homework help 
Diagram Generator, Flashcard Maker, Notes Generator, Research Assistant, Answer Generator, AI Homework Helper & AI Detector