Certainly! Let's find the derivative of the function [tex]\( y = a^x \)[/tex] with respect to [tex]\( x \)[/tex]. Here, [tex]\( a \)[/tex] is a positive constant.
1. Expression of the Function:
[tex]\[
y = a^x
\][/tex]
2. Using the Chain Rule:
To differentiate [tex]\( y = a^x \)[/tex] with respect to [tex]\( x \)[/tex], we can use the chain rule involving the natural logarithm function. Recall that [tex]\( a^x \)[/tex] can be expressed using the natural exponential function:
[tex]\[
a^x = e^{x \ln a}
\][/tex]
3. Differentiate Using the Chain Rule:
Now, let’s differentiate:
[tex]\[
\frac{d}{dx} \left( e^{x \ln a} \right)
\][/tex]
The derivative of [tex]\( e^{u} \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( e^u \cdot \frac{du}{dx} \)[/tex]. Here, [tex]\( u = x \ln a \)[/tex].
4. Compute [tex]\( \frac{du}{dx} \)[/tex]:
[tex]\[
\frac{du}{dx} = \frac{d}{dx} (x \ln a) = \ln a
\][/tex]
5. Apply the Chain Rule:
Putting it all together:
[tex]\[
\frac{d}{dx} \left( e^{x \ln a} \right) = e^{x \ln a} \cdot \ln a
\][/tex]
Recall that [tex]\( e^{x \ln a} = a^x \)[/tex]. Therefore,
[tex]\[
\frac{d}{dx} (a^x) = a^x \cdot \ln a
\][/tex]
6. Conclusion:
Thus, the derivative of [tex]\( y = a^x \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{dy}{dx} = a^x \ln a
\][/tex]
This concludes our verification that:
[tex]\[
\frac{dy}{dx} = a^x \ln a
\][/tex]
