Q. \[ \frac{d}{dx}\left(8^x\right) \]

Q: . What is the derivative of 8^x?

. \dfrac{d}{dx}(8^x)=8^x\ln(8).

Q: . How do you differentiate a^x in general?

. For a>0, a\neq 1: \dfrac{d}{dx}(a^x)=a^x\ln(a).

Q: . What if the base is 2 instead, like 2^x?

. \dfrac{d}{dx}(2^x)=2^x\ln(2).

Q: . How do you differentiate 8^{x+3}?

. Treat u=x+3. Then \dfrac{d}{dx}(8^{x+3})=8^{x+3}\ln(8).

Q: . How do you differentiate 8^{3x}?

. Use u=3x. Then \dfrac{d}{dx}(8^{3x})=8^{3x}\ln(8)\cdot 3.

Q: . Why does the \ln(8) appear?

. Because 8^x=e^{x\ln(8)}. Then \dfrac{d}{dx}(e^{x\ln(8)})=e^{x\ln(8)}\ln(8)=8^x\ln(8).

Answer

Use the rule \(\frac{d}{dx}\left(a^x\right)=a^x\ln(a)\). Here \(a=8\).

\[
\frac{d}{dx}\left(8^x\right)=8^x\ln(8)
\]

Detailed Explanation

We want to find the derivative of the function \(f(x)=8^x\).

Step 1: Identify the form

The function is an exponential with a variable in the exponent: \(8^x\). This matches the general rule for bases greater than zero and not equal to \(1\):

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

Step 2: Substitute the base

Here, \(a=8\). Substitute into the rule:

\[
\frac{d}{dx}\left(8^x\right)=8^x\ln(8).
\]

Step 3: Write the final derivative

Therefore, the derivative is:

\[
\boxed{\\frac{d}{dx}\left(8^x\right)=8^x\ln(8)}.
\]

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Calculus FAQ

. What is the derivative of \(8^x\)?

. \(\dfrac{d}{dx}(8^x)=8^x\ln(8)\).

. How do you differentiate \(a^x\) in general?

. For \(a>0\), \(a\neq 1\): \(\dfrac{d}{dx}(a^x)=a^x\ln(a)\).

. What if the base is \(2\) instead, like \(2^x\)?

. \(\dfrac{d}{dx}(2^x)=2^x\ln(2)\).

. How do you differentiate \(8^{x+3}\)?

. Treat \(u=x+3\). Then \(\dfrac{d}{dx}(8^{x+3})=8^{x+3}\ln(8)\).

. How do you differentiate \(8^{3x}\)?

. Use \(u=3x\). Then \(\dfrac{d}{dx}(8^{3x})=8^{3x}\ln(8)\cdot 3\).

. Why does the \(\ln(8)\) appear?

. Because \(8^x=e^{x\ln(8)}\). Then \(\dfrac{d}{dx}(e^{x\ln(8)})=e^{x\ln(8)}\ln(8)=8^x\ln(8)\).

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