Q. \[ \frac{d}{dx}\left(6^x\right)=6^x\ln(6) \]

Q: What is the derivative of 6^x?

\dfrac{d}{dx}\left(6^x\right)=6^x\ln(6).

Q: How do you differentiate a^x for a>0, a\neq 1?

\dfrac{d}{dx}\left(a^x\right)=a^x\ln(a).

Q: How do you differentiate 6^{2x}?

\dfrac{d}{dx}\left(6^{2x}\right)=2\cdot 6^{2x}\ln(6).

Q: What is \dfrac{d}{dx}\left(6^{x+3}\right)?

\dfrac{d}{dx}\left(6^{x+3}\right)=6^{x+3}\ln(6).

Q: What is \dfrac{d}{dx}\left(3\cdot 6^x\right)?

\dfrac{d}{dx}\left(3\cdot 6^x\right)=3\cdot 6^x\ln(6).

Q: How do you differentiate 6^{-x}?

\dfrac{d}{dx}\left(6^{-x}\right)=-6^{-x}\ln(6).

Answer

Rewrite \(6^x\) in terms of the exponential rule: \(\dfrac{d}{dx} a^x = a^x \ln(a)\) for \(a>0\), \(a\neq 1\).

With \(a=6\):

\[
\frac{d}{dx}6^x = 6^x \ln(6)
\]

Detailed Explanation

We want to find the derivative of the function

\[
f(x)=6^x.
\]

Step 1: Identify the function type.

The expression \(6^x\) is an exponential function where the base \(6\) is constant and the exponent is \(x\). A key fact is that the derivative of \(a^x\) (with \(a>0\) and \(a\neq 1\)) is related to the natural logarithm of \(a\).

Step 2: Use the standard derivative rule.

For \(a>0\) and \(a\neq 1\), the derivative rule is

\[
\frac{d}{dx}\left(a^x\right)=a^x\ln(a).
\]

Step 3: Apply the rule with \(a=6\).

Substitute \(a=6\) into the formula. Then \(a^x\) becomes \(6^x\), and \(\ln(a)\) becomes \(\ln(6)\).

\[
\frac{d}{dx}\left(6^x\right)=6^x\ln(6).
\]

Final answer:

\[
\boxed{\frac{d}{dx}\left(6^x\right)=6^x\ln(6).}
\]

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Calculus FAQ

What is the derivative of \(6^x\)?

\(\dfrac{d}{dx}\left(6^x\right)=6^x\ln(6)\).

How do you differentiate \(a^x\) for \(a>0\), \(a\neq 1\)?

\(\dfrac{d}{dx}\left(a^x\right)=a^x\ln(a)\).

How do you differentiate \(6^{2x}\)?

\(\dfrac{d}{dx}\left(6^{2x}\right)=2\cdot 6^{2x}\ln(6)\).

What is \(\dfrac{d}{dx}\left(6^{x+3}\right)\)?

\(\dfrac{d}{dx}\left(6^{x+3}\right)=6^{x+3}\ln(6)\).

What is \(\dfrac{d}{dx}\left(3\cdot 6^x\right)\)?

\(\dfrac{d}{dx}\left(3\cdot 6^x\right)=3\cdot 6^x\ln(6)\).

How do you differentiate \(6^{-x}\)?

\(\dfrac{d}{dx}\left(6^{-x}\right)=-6^{-x}\ln(6)\).

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