Lesson 15

Derivative of the inverse of a function. in order to calculate (f −1). ▷ consider y = f −1(x) so then f (y) = x; write instead.
Full Text
Lesson 15Read 3.5, 3.6Logarithmic differentiationDerivatives of inverse functionsformulas to knowI (ln x)′= 1xI (arctan x)′= 11+x2I (arcsin x)′= 1√1−x2I (arccos x)′= − 1√1−x2Derivative of the inverse of a functionIn order to calculate (f −1)′I Consider y = f −1(x) so then f (y ) = x; write insteady = y (x) , and f (y (x)) = x . (*) .I Take the derivative with respect to x of both sides of (*).I You get dfdydydx = 1 .I You get dydx (x) = 1dfdy (y ) . Simplify and get a formula in xExample. Derive the formula (arcsin x)′= 1√1−x2Example. Derive the formula (ln x)′= 1xCalculate the derivative of ln(|x|)Calculate the derivative of ln(3x + ln(3x + ln(3x))))Logarithmic differentiationThis method is used to find derivatives of functions of the formg (x)h(x)I Write y = g (x)h(x) and take ln of both sides.I You get ln(y ) = h(x) ln(g (x)) . Now differentiate both sideswith respect to x.I You get 1y y ′= h′ln g + h g ′g . Solve for y ′.I You get y ′= y (h′ln g ′+ h g ′g ). Replace y with g (x)h(x...