Take a multiple choice test on derivatives.
In this test you will be asked questions about the following topics:
Derivative of a logarithmic function
Derivative of an exponential function
Derivative of a linear combination of functions
When you give a right answer, it is marked with .
When you give a wrong answer, it is marked with and the right answer is marked with .
When you answer a question, your score is updated and you can read an explanation of the right answer before going to the next question.
Are you ready?
Question 1. DefineThe derivative of is:
Explanation. The derivative of the logarithmic function is:See the calculus review page entitled Derivatives - Review.
Question 2. DefineThe second derivative of is:
Explanation. The first derivative of the logarithmic function is:Therefore, the second derivative of is:Using the formula for the derivative of a power function, we obtain:See the calculus review page entitled Derivatives - Review.
Question 3. DefineThe derivative of is:
Explanation. The derivative of a generic logarithmic function is:Here . Therefore:See the calculus review page entitled Derivatives - Review.
Question 4. DefineThe second derivative of is:
Explanation. The first derivative of the logarithmic function is:Therefore, the second derivative of is:Note that we have been able to use the linearity of the derivative because is a constant. See the calculus review page entitled Derivatives - Review.
Question 5. DefineThe derivative of is:
Explanation. is a constant. So, by the linearity of the derivative:Now, the rule for differentiating exponential functions is:Therefore:See the calculus review page entitled Derivatives - Review.
Question 6. DefineThe second derivative of is:
Explanation. The first derivative of the exponential function is:Therefore, the second derivative of is:Note that we have been able to use the linearity of the derivative because is a constant. See the calculus review page entitled Derivatives - Review.
Question 7. DefineThe derivative of is:
Explanation. By linearity of the derivative:The first summand is: because the derivative of a constant is . The second summand is:by the rule for differentiating exponentials. Therefore:See the calculus review page entitled Derivatives - Review.
Question 8. DefineThe derivative of is:
Explanation. The derivative of is:Note that we have been able to use the linearity of the derivative because is a constant. See the calculus review page entitled Derivatives - Review.
Question 9. The function:is the derivative of:
Explanation. By linearity:See the calculus review page entitled Derivatives - Review.
Question 10. The function:is the second derivative of:
Explanation. The first derivative of the functionis:The second derivative is:See the calculus review page entitled Derivatives - Review.
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