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

We want to find the derivative of the function \(y=b^{x}\), where \(b\) is a constant (and \(b>0\), \(b\neq 1\)).

Step 1: Recall the exponential differentiation rule

Exponential functions like \(e^{x}\) are special because their derivative is the same as the original function: \(\frac{d}{dx}e^{x}=e^{x}\).

To differentiate \(b^{x}\), we rewrite it using the natural exponential function \(e^{x}\).

Step 2: Rewrite \(b^{x}\) in terms of \(e\)

Use the identity

\[
b^{x}=e^{x\ln(b)}
\]

Explanation: \(\ln(b)\) is the natural logarithm of \(b\), and this identity holds because exponentials and logarithms are inverses.

Step 3: Differentiate using the chain rule

Now let

\[
y=e^{x\ln(b)}
\]

The chain rule says:

\[
\frac{d}{dx}e^{u}=e^{u}\cdot \frac{du}{dx}
\]

Here \(u=x\ln(b)\). Compute \(\frac{du}{dx}\):

\[
u=x\ln(b)\quad \Rightarrow \quad \frac{du}{dx}=\ln(b)
\]

Now apply the chain rule:

\[
\frac{dy}{dx}=e^{x\ln(b)}\cdot \ln(b)
\]

Step 4: Convert back to \(b^{x}\)

Since \(e^{x\ln(b)}=b^{x}\), we get

\[
\frac{dy}{dx}=b^{x}\ln(b)
\]

Final Answer

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

This is the derivative of \(b^{x}\) with respect to \(x\).