Question

Find the derivative...PLEASE HELP!!
equation in comments

Answer 1

\[\frac{ d }{ dz }\int\limits_{z}^{\pi} \cos(t ^{3})dt\]

Answer 2

This is actually a trick of integration here. This fits the model as follows: \[\frac{ d }{ dx } \int\limits_{x}^{k}f(t)dt\] where t is the variable of integration, x is a variable, and k is a constant. From the operation of integration, we will obtain something that follows: \[\frac{ d }{ dx }\left[ F(k) - F(x) \right]\]In this scenario, you have the integrated function of f(t), given by the notation F(t), where k is a constant and x is a new variable supplied by the parameters of integration. Realize that plugging in k, a constant, into the function F(t) is the same as plugging in any constant into a function -- it gets you some numerical value, some other constant. So this actually changes the above integrated statement to: \[\frac{ d }{ dx }\left[ C - F(x) \right]\]You can rewrite this as: \[\frac{ d }{ dx }(C) - \frac{ d }{ dx }\left[ F(x) \right]\]Laws of derivatives state that the derivative respective to a variable of a constant is just zero. And since we know that F(t) is the higher order antiderivative of f(t), the derivative of F(t) would just become f(t). Apply both those concepts here to obtain the final answer: \[0 - f(x)\] This, of course, is rewritten to get the answer: -f(x). Of course, this is all in terms of the variables I used, so work with your variables to get your answer.

Answer 3

ok, so would the answer be \[-\cos(t ^{3})\]

Answer 4

Be careful here. The term in the initial integrand is t^3, not just t. Usually that means that you have to consider it like the chain rule, where you have to apply derivation/integration techniques to not just the function but also the variable term.

Answer 5

so \[-3t ^{2} \cos(t ^{3})\] but substituting z for t?

Answer 6

actually, forget that, it would just be my first answer since the derivative of x is 1

Answer 7

my first answer was correct