Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

So \[
g'(x) = -4\tan(x)
\]