Q. \(\frac{d}{dx}\left(5^x\right)\)
Answer
Use the rule \(\frac{d}{dx}a^x=a^x\ln(a)\). Here \(a=5\), so
\[
\frac{d}{dx}\left(5^x\right)=5^x\ln(5).
\]
Detailed Explanation
We want to find the derivative of the function \(f(x)=5^x\).
Step 1: Identify the function type
The function \(5^x\) is an exponential function with a constant base \(5\). A key rule for differentiating exponential functions of the form \(a^x\) is used.
Step 2: Use the differentiation rule
If \(a>0\) and \(a\neq 1\), then
\[
\frac{d}{dx}\left(a^x\right)=a^x\ln(a).
\]
Step 3: Substitute \(a=5\)
Here \(a=5\), so we substitute into the rule:
\[
\frac{d}{dx}\left(5^x\right)=5^x\ln(5).
\]
Final Answer
\[
\frac{d}{dx}\left(5^x\right)=5^x\ln(5).
\]
See full solution
Calculus FAQ
What is the derivative of \(5^x\)?
\(\dfrac{d}{dx}[5^x]=5^x\ln(5)\).
What is the derivative of \(a^x\) for \(a>0\), \(a\neq 1\)?
\(\dfrac{d}{dx}[a^x]=a^x\ln(a)\).
How do you differentiate \(5^{2x}\)?
Use the chain rule: \(\dfrac{d}{dx}[5^{2x}]=5^{2x}\ln(5)\cdot 2=2\ln(5)\,5^{2x}\).
What is the derivative of \(\sqrt{5^x}\)?
\(\sqrt{5^x}=(5^x)^{1/2}=5^{x/2}\). Then \(\dfrac{d}{dx}[5^{x/2}]=5^{x/2}\ln(5)\cdot \dfrac{1}{2}=\dfrac{\ln(5)}{2}5^{x/2}\).
Differentiate \(e^{x\ln 5}\) and relate it to \(5^x\).
Since \(5^x=e^{x\ln 5}\), \(\dfrac{d}{dx}[e^{x\ln 5}]=e^{x\ln 5}\cdot \ln 5=5^x\ln 5\).
What is \(\dfrac{d}{dx}[5^x+3]\)?
The constant derivative is \(0\). So \(\dfrac{d}{dx}[5^x+3]=5^x\ln(5)\).
What is the derivative of \(5^{-x}\)?
\(5^{-x}=(5^x)^{-1}\). Use the rule \( \dfrac{d}{dx}[a^{kx}]=a^{kx}\ln(a)\cdot k\). So \(\dfrac{d}{dx}[5^{-x}]=5^{-x}\ln(5)\cdot(-1)=-5^{-x}\ln(5)\).
Use these tools for 5^x.
Get math help step by step.
Get math help step by step.
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