How do you use the Fundamental Theorem of Calculus to find the derivative of #int (cos(t^4) + t) dt# from -4 to sinx?

1 Answer
Oct 19, 2017

Answer:

Use the Chain Rule along with the Fundamental Theorem of Calculus to get the answer #(cos(sin^{4}(x))+sin(x))*cos(x)#

Explanation:

For a function #F(x)=int_{a}^{x}f(t)\ dt#, the Fundamental Theorem of Calculus implies that, if #f# is continuous, then #F'(x)=f(x)#.

If #f(t)=cos(t^{4})+t#, the function to differentiate in this problem is #h(x)=int_{-4}^{sin(x)}f(t)\ dt#. If #F(x)=int_{-4}^{x}f(t)\ dt#, then #h(x)=F(sin(x))#.

By the Chain Rule, #h'(x)=F'(sin(x)) * d/dx(sin(x))#

#=(cos(sin^{4}(x))+sin(x)) * cos(x)#.

(In general, the Chain Rule says that, under appropriate assumptions about differentiability, #d/dx(f(g(x)))=f'(g(x))*g'(x)#).