Math Problem Statement
solve
Solution
To differentiate the given function , we will apply the chain rule and the derivative of the logarithmic function. Here are the steps:
-
Recall the derivative of :
If , the derivative is given by: where is the inner function, and is the base of the logarithm (in this case, 10). -
Identify :
For this function, . -
Differentiate :
The derivative of is: -
Apply the chain rule:
Using the formula for the derivative of a logarithmic function, we get:
Thus, the derivative of the function is:
Let me know if you need further clarification! Would you like more details on any step?
Related Questions:
- How would the derivative change if the logarithm had a different base?
- What happens if the inner function was more complex, such as ?
- How does the chain rule apply to other types of functions like exponentials?
- Can you explain the role of in this derivative?
- How would this problem change if the logarithm was natural, ?
Tip:
Always make sure to identify the inner function clearly when using the chain rule for differentiation!
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Math Problem Analysis
Mathematical Concepts
Calculus
Logarithmic Differentiation
Chain Rule
Formulas
Derivative of log_b(u(x)): f'(x) = (1 / (u(x) * ln(b))) * u'(x)
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12, College-level Calculus
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