Problem 29 Differentiate. $$g(x)=\frac{7^... [FREE SOLUTION] (2024)

Short Answer

exponential functions

Exponential functions are a special type of function where a constant base is raised to a variable exponent. In the function provided in the exercise, the exponential part is represented by \( 7^x \).
When differentiating an exponential function like \( a^x \) (where \( a \) is a constant), the derivative is \( a^x \, \text{ln} \, a \). This means that for \( 7^x \), the derivative is \( 7^x \, \text{ln} \, 7 \).
This property is used in step 3 of our solution to find the derivative of the numerator.

logarithmic differentiation

Logarithmic differentiation is a technique used to differentiate functions by using the natural logarithm. It's especially helpful when dealing with complicated products, quotients, or exponentials.
In this exercise, although we didn’t explicitly take the logarithm of the entire function, we understand that for exponential functions, knowing the properties of logarithms is crucial.
For example, when differentiating \( 7^x \), we directly use the property of logarithms to get the derivative: \( 7^x \, \text{ln} \, 7 \).
Understanding logarithmic differentiation means recognizing relationships like \( \text{ln}(a^x) = x \, \text{ln}(a) \). Even though we don’t apply this exact relationship here, it forms the basis for why differentiating exponentials involves a natural logarithm.

derivative