Q. \(\dfrac{d}{dx}\left(n^{x}\right)\)

Q: What is the derivative of n^x (with constant n>0, n\neq 1)?

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

Q: How do you derive \dfrac{d}{dx}\left(n^x\right) using a^x=e^{x\ln a}?

Write n^x=e^{x\ln n}. Then \dfrac{d}{dx}e^{x\ln n}=e^{x\ln n}(\ln n)=n^x\ln(n).

Q: What if the base is e? What is \dfrac{d}{dx}(e^x)?

Since \ln(e)=1, \dfrac{d}{dx}(e^x)=e^x\ln(e)=e^x.

Q: What is \dfrac{d}{dx}\left(5^{x}\right)?

\dfrac{d}{dx}\left(5^x\right)=5^x\ln(5).

Q: What is \dfrac{d}{dx}\left( (1/2)^{x}\right)?

\dfrac{d}{dx}\left((1/2)^x\right)=(1/2)^x\ln(1/2). Note \ln(1/2)=-\ln 2.

Q: Does the formula work for n\le 0?

For real-valued derivatives, require n>0, n\neq 1 so \ln(n) is real. Then \dfrac{d}{dx}(n^x)=n^x\ln(n).

Answer

To differentiate \(n^x\) (with constant \(n>0\) and \(n\neq 1\)), rewrite using \(n^x=e^{x\ln n}\). Then use the chain rule.

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

Final answer: \(\frac{d}{dx}\left(n^x\right)=n^x\ln n\).

Detailed Explanation

We want to find the derivative of the function \(f(x)=n^x\), where \(n\) is a constant (with \(n>0\) and \(n\neq 1\)).

Step 1: Recall how to differentiate an exponential with base \(n\)

A standard result is: if \(f(x)=a^x\) where \(a>0\) and \(a\neq 1\), then

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

Here, the base is \(a=n\).

Step 2: Apply the formula with \(a=n\)

Substitute \(a=n\) into the derivative rule:

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

Final Answer

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

See full solution

Get AI homework help, try it now.

AI Homework Helper

Calculus FAQ

What is the derivative of \(n^x\) (with constant \(n>0\), \(n\neq 1\))?

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

How do you derive \( \dfrac{d}{dx}\left(n^x\right)\) using \(a^x=e^{x\ln a}\)?

Write \(n^x=e^{x\ln n}\). Then \(\dfrac{d}{dx}e^{x\ln n}=e^{x\ln n}(\ln n)=n^x\ln(n)\).

What if the base is \(e\)? What is \(\dfrac{d}{dx}(e^x)\)?

Since \(\ln(e)=1\), \(\dfrac{d}{dx}(e^x)=e^x\ln(e)=e^x\).

What is \(\dfrac{d}{dx}\left(5^{x}\right)\)?

\(\dfrac{d}{dx}\left(5^x\right)=5^x\ln(5)\).

What is the derivative of \(n^{u(x)}\) (chain rule)?

\(\dfrac{d}{dx}\left(n^{u(x)}\right)=n^{u(x)}\ln(n)\cdot u'(x)\).

What is \(\dfrac{d}{dx}\left( (1/2)^{x}\right)\)?

\(\dfrac{d}{dx}\left((1/2)^x\right)=(1/2)^x\ln(1/2)\). Note \(\ln(1/2)=-\ln 2\).

Does the formula work for \(n\le 0\)?

For real-valued derivatives, require \(n>0\), \(n\neq 1\) so \(\ln(n)\) is real. Then \(\dfrac{d}{dx}(n^x)=n^x\ln(n)\).

Use AI tools to learn this derivative. Solve dy/dx of n^x faster.

298,376+ active customers
Math, Geometry, Trigonometry, etc.

AI Math Solver Geometry AI Trig Solver