To find the derivative of the function [tex]\( f(x) = 12^{3x} \)[/tex], we will use the chain rule and the properties of logarithms. Here's a step-by-step solution:
1. Rewrite the Expression:
First, recognize that the function [tex]\( 12^{3x} \)[/tex] can be rewritten using properties of exponents. However, it's more straightforward to directly apply the chain rule to the original form.
2. Differentiate Using the Chain Rule:
The chain rule states that if you have a composite function [tex]\( f(g(x)) \)[/tex], then its derivative [tex]\( \frac{d}{dx}(f(g(x))) \)[/tex] is [tex]\( f'(g(x)) \cdot g'(x) \)[/tex].
In this case, let [tex]\( u = 3x \)[/tex]. So, [tex]\( f(u) = 12^u \)[/tex]. The derivative of [tex]\( 12^u \)[/tex] with respect to [tex]\( u \)[/tex] is given by:
[tex]\[
\frac{d}{du} \left(12^u \right) = 12^u \ln(12)
\][/tex]
Now, we need to multiply this by the derivative of [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{d}{dx} (3x) = 3
\][/tex]
3. Apply the Chain Rule:
Combining these results, we get:
[tex]\[
\frac{d}{dx} \left(12^{3x}\right) = 12^{3x} \ln(12) \cdot 3
\][/tex]
4. Simplify the Expression:
Finally, multiply the terms together:
[tex]\[
3 \cdot 12^{3x} \ln(12)
\][/tex]
Therefore, the derivative of the function [tex]\( 12^{3x} \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\operatorname{Dx}\left(12^{3x}\right) = 3 \cdot 12^{3x} \ln(12)
\][/tex]
