Math Problem Statement

Differentiate various functions on a derivative worksheet, including examples like f(x) = 4x^5 - 5x^4, f(x) = e^x sin(x), f(x) = ln(5x^2 + 9)^3, etc.

Solution

It looks like you've uploaded another image of the same derivatives worksheet. Is there a specific problem you'd like help with from this sheet? Let me know the number of the problem you'd like to work on, and I can guide you through the solution.

If you'd like help with a particular type of derivative (e.g., chain rule, product rule, implicit differentiation), feel free to mention that as well!

Here are a few specific suggestions:

  1. Would you like me to solve problem 4 (which involves the chain rule)?
  2. Need assistance with problem 8 (which seems to be a product or chain rule problem)?
  3. Want help on implicit differentiation, like in problem 41?
  4. Should I walk you through how to differentiate a natural logarithmic function, such as in problem 9 or problem 10?
  5. Interested in problem 21 for dealing with fractional exponents and exponentials?

Tip: In differentiation problems involving exponentials like exe^x, keep in mind that the derivative of eue^u is eududxe^u \cdot \frac{du}{dx}, where uu is a function of xx.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Calculus
Chain Rule
Product Rule
Quotient Rule
Implicit Differentiation

Formulas

Derivative of e^x is e^x
Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x)
Product Rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
Quotient Rule: (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x))/g(x)^2

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12, College Level