There are a few common reasons for this. One possibility is that you may be making a computational error, such as forgetting to apply the chain rule or misplacing a negative sign. Another possibility is that the function you are trying to differentiate is not continuous or differentiable at the point you are interested in. Lastly, it is possible that the expected derivative is incorrect. It is always a good idea to double check your work and consult with a teacher or tutor if you are still unsure.
When dealing with a complicated function, it is important to break it down into smaller, more manageable pieces. Use the chain rule, product rule, and quotient rule as needed. Simplify as much as possible before taking the derivative. It may also be helpful to use tables or graphs to visualize the function and its derivative.
To determine whether a critical point is a maximum or minimum, you can use the second derivative test. Take the second derivative of the function and plug in the critical point. If the resulting value is positive, the critical point is a minimum. If the resulting value is negative, the critical point is a maximum. If the resulting value is zero, the test is inconclusive and you may need to use other methods, such as graphing, to determine the nature of the critical point.
Yes, you can use the chain rule to find the derivative of an implicit function. First, rewrite the implicit function as an explicit function by solving for the dependent variable. Then, use the chain rule as you normally would for an explicit function. Remember to also apply the chain rule when taking the derivative of the dependent variable with respect to the independent variable.
To find the absolute maximum or minimum of a function, you can use the first or second derivative test, depending on the function. You can also use the Extreme Value Theorem, which states that a continuous function on a closed interval will have both an absolute maximum and minimum. Set the derivative equal to zero and solve for critical points within the interval, and then evaluate the function at each critical point and at the endpoints of the interval to determine the absolute maximum and minimum.