Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.6 Logarithmic And Other Functions Defined By Integrals - Exercises Set 6.6 - Page 460: 24

$$\eqalign{ & \left( {\text{a}} \right){\text{Proved}} \cr & \left( {\text{b}} \right){\text{Proved}} \cr} $$

$$\eqalign{ & \left( {\text{a}} \right)\frac{d}{{dx}}\left[ {\int_x^a {f\left( t \right)dt} } \right] = - f\left( x \right) \cr & {\text{Apply the property }}\int_a^b {f\left( x \right)} dx = - \int_b^a {f\left( x \right)} dx \cr & \frac{d}{{dx}}\left[ {\int_x^a {f\left( t \right)dt} } \right] = \frac{d}{{dx}}\left[ { - \int_a^x {f\left( t \right)dt} } \right] = - \frac{d}{{dx}}\left[ {\int_a^x {f\left( t \right)dt} } \right] \cr & {\text{Where }}\frac{d}{{dx}}\left[ {\int_a^x {f\left( t \right)dt} } \right] = f\left( x \right),{\text{ then}} \cr & - \frac{d}{{dx}}\left[ {\int_a^x {f\left( t \right)dt} } \right] = - f\left( x \right) \cr & \cr & \left( {\text{b}} \right)\frac{d}{{dx}}\left[ {\int_{g\left( x \right)}^a {f\left( t \right)dt} } \right] = - f\left( {g\left( x \right)} \right)g'\left( x \right) \cr & {\text{Apply the property }}\int_a^b {f\left( x \right)} dx = - \int_b^a {f\left( x \right)} dx \cr & \frac{d}{{dx}}\left[ {\int_{g\left( x \right)}^a {f\left( t \right)dt} } \right] = \frac{d}{{dx}}\left[ { - \int_a^{g\left( x \right)} {f\left( t \right)dt} } \right] \cr & = - \frac{d}{{dx}}\left[ {\int_a^{g\left( x \right)} {f\left( t \right)dt} } \right] \cr & {\text{By the chain rule}} \cr & - \frac{d}{{dx}}\left[ {\int_a^{g\left( x \right)} {f\left( t \right)dt} } \right] = - \left[ {f\left( {g\left( x \right)} \right)g'\left( x \right)} \right] \cr & - \frac{d}{{dx}}\left[ {\int_a^{g\left( x \right)} {f\left( t \right)dt} } \right] = - f\left( {g\left( x \right)} \right)g'\left( x \right) \cr} $$