Math Problem Statement

solve

Differentiate the function f(x) = log_10(x^3 + 2)

Solution

To differentiate the given function f(x)=log10(x3+2)f(x) = \log_{10}(x^3 + 2), we will apply the chain rule and the derivative of the logarithmic function. Here are the steps:

  1. Recall the derivative of logb(u)\log_b(u):
    If f(x)=logb(u(x))f(x) = \log_b(u(x)), the derivative is given by: f(x)=1u(x)ln(b)u(x)f'(x) = \frac{1}{u(x) \ln(b)} \cdot u'(x) where u(x)u(x) is the inner function, and bb is the base of the logarithm (in this case, 10).

  2. Identify u(x)u(x):
    For this function, u(x)=x3+2u(x) = x^3 + 2.

  3. Differentiate u(x)u(x):
    The derivative of u(x)=x3+2u(x) = x^3 + 2 is: u(x)=3x2u'(x) = 3x^2

  4. Apply the chain rule:
    Using the formula for the derivative of a logarithmic function, we get: f(x)=1(x3+2)ln(10)3x2f'(x) = \frac{1}{(x^3 + 2) \ln(10)} \cdot 3x^2

Thus, the derivative of the function is: f(x)=3x2(x3+2)ln(10)f'(x) = \frac{3x^2}{(x^3 + 2) \ln(10)}

Let me know if you need further clarification! Would you like more details on any step?

Related Questions:

  1. How would the derivative change if the logarithm had a different base?
  2. What happens if the inner function was more complex, such as x4+3x2+1x^4 + 3x^2 + 1?
  3. How does the chain rule apply to other types of functions like exponentials?
  4. Can you explain the role of ln(10)\ln(10) in this derivative?
  5. How would this problem change if the logarithm was natural, ln(x3+2)\ln(x^3 + 2)?

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