Question

Find \(z'(x)\)

Answer 1

If \(z(x)=\int_{x}^{2} ln(t)~ dt\) then find \(z'(t)\)

Answer 2

\(z'(x)\)

Answer 3

Well, if you substitute \(t=x\) then \(z'(t)=z'(x)\).

Answer 4

\(\large z(x)(\frac{d}{dx})=(\frac{d}{dx})\int_{2}^{x} ln t~dt=z'(x)=-ln~x\)?

Answer 5

*\(= - (\frac{d}{dx})\int_{2}^{x} ln t~dt\)

Answer 6

\((\int_x^2 ln(t)\ dt)' = \left(\int_0^2\ln(t)\ dt - \int_0^x \ln(t)\ dt\right)'\)
Now apply the fundamental theorem to each term and
\((\int_x^2 ln(t)\ dt)' = \left(\int_0^2\ln(t)\ dt - \int_0^x \ln(t)\ dt\right)' = \left(c - \int_0^x \ln(t)\ dt\right)' = 0 - \ln(x)\)

Answer 7

Can't wait to learn this..