Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function g(x) = 4tan(t)dt integral x to 7

Question

anonymous · Answer

$$
g(x)=\int \limits_x^7 4\tan(t)\;dt
$$We want to find $g'(x)$?

anonymous · Answer

Consider the function $f(t)$ such that $f'(t) = 4\tan(t)$.
This gives us: $$
g(x) =\int\limits_x^7f'(t)\;dt= f(7)-f(x)
$$

anonymous · Answer

Now looking at $$
g(x)=f(7)-f(x)
$$We can differentiate both sides to get: $$
g'(x) = 0-f'(x)
$$

anonymous · Answer

Remember that $f'(x) = f'(t)\Bigg|^{t=x}= 4\tan(t)\Bigg|^{t=x}= 4\tan(x)$.

anonymous · Answer

So $$
g'(x) = -4\tan(x)
$$