To find the derivative of the function [tex]\( g(x) = 6x e^{7x} \)[/tex], we will use the product rule. The product rule states that if we have a function in the form [tex]\( u(x) \cdot v(x) \)[/tex], then its derivative is given by:
[tex]\[
(uv)' = u'v + uv'
\][/tex]
For [tex]\( g(x) = 6x e^{7x} \)[/tex], we can identify [tex]\( u(x) = 6x \)[/tex] and [tex]\( v(x) = e^{7x} \)[/tex].
Now, let's find the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex] individually:
1. Derivative of [tex]\( u(x) = 6x \)[/tex]:
[tex]\[ u'(x) = 6 \][/tex]
2. Derivative of [tex]\( v(x) = e^{7x} \)[/tex]:
Here, we need to use the chain rule, as [tex]\( v(x) = e^{7x} \)[/tex] is a composition of functions. Let [tex]\( h(x) = 7x \)[/tex]. Then the derivative of [tex]\( e^{h(x)} \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d}{dx} (e^{7x}) = e^{7x} \cdot 7 = 7 e^{7x}
\][/tex]
Now, applying the product rule:
[tex]\[
g'(x) = u'(x)v(x) + u(x)v'(x)
\][/tex]
Substituting the values we calculated:
[tex]\[
g'(x) = 6 \cdot e^{7x} + 6x \cdot 7 e^{7x}
\][/tex]
[tex]\[
g'(x) = 6e^{7x} + 42x e^{7x}
\][/tex]
Therefore, the derivative [tex]\( g'(x) \)[/tex] is:
[tex]\[
g'(x) = 42x e^{7x} + 6 e^{7x}
\][/tex]
