To solve the problem of finding the derivative of the function [tex]\( g(x) = 7^x + 1 \)[/tex], let's go through the process step-by-step.
### Step-by-Step Solution
1. Identify the Function: We start with the function [tex]\( g(x) = 7^x + 1 \)[/tex].
2. Understand the Terms:
- The term [tex]\( 7^x \)[/tex] is an exponential function.
- The constant term [tex]\( +1 \)[/tex] has no dependence on [tex]\( x \)[/tex] and its derivative will be zero.
3. Apply the Derivative Rules:
- We need to find the derivative of [tex]\( 7^x \)[/tex].
- Recall the derivative rule for exponential functions [tex]\( a^x \)[/tex]. The derivative of [tex]\( a^x \)[/tex] where [tex]\( a \)[/tex] is a constant, is given by:
[tex]\[
\frac{d}{dx} a^x = a^x \ln(a)
\][/tex]
Here, [tex]\( a = 7 \)[/tex], so the derivative of [tex]\( 7^x \)[/tex] is:
[tex]\[
\frac{d}{dx} 7^x = 7^x \ln(7)
\][/tex]
4. Differentiate Each Term:
- Differentiate the exponential term [tex]\( 7^x \)[/tex]:
[tex]\[
\frac{d}{dx} 7^x = 7^x \ln(7)
\][/tex]
- Differentiate the constant term [tex]\( +1 \)[/tex]:
[tex]\[
\frac{d}{dx} 1 = 0
\][/tex]
5. Combine Derivatives:
- Combine the derivatives of each term:
[tex]\[
g'(x) = 7^x \ln(7) + 0
\][/tex]
- Simplifying this, we get:
[tex]\[
g'(x) = 7^x \ln(7)
\][/tex]
### Result:
The derivative of the function [tex]\( g(x) = 7^x + 1 \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
g'(x) = 7^x \ln(7)
\][/tex]
This is the final answer.
