\(In\,\,The\,\,Case\,\,Of\,\,Nested\,\,Functions:\)
\(\dfrac{dy}{dx} = [\dfrac{du}{dx}][\dfrac{dy}{du}]\)
\(ex\,\,a.\)
\(u = x^{2} - 4\)
\(\dfrac{du}{dx} = 2x\)
\(\dfrac{dy}{dx} = (u)^{3}\)
\(The\,\,Main\,\,Problem:\)
\(\dfrac{dy}{dx} = (x^{2} - 4)^{3} \)
\([\dfrac{du}{dx}][\dfrac{dy}{du}] =\)
\([2x][(3)(x^{2} - 4)^{3 - 1}] = \)
\(6x(x^{2} - 4)^{2} = \)
\(6x(x^{2} - 4)(x^{2} - 4) =\)
\(6x[2x^{2} - 4x^{2} - 4x^{2} + 16] = \)
\(6x[x^{4} - 8x^{2} + 16] =\)
\(6x^{5} - 48x^{3} + 96x\)
\(ex. b\)
\(u = 4x\)
\(\dfrac{du}{dx} = 4\)
\(\dfrac{dy}{du} = \cos(u) \)
\(The\,\,Main\,\,Problem\)
\(\dfrac{dy}{dx} = \sin(4x)\)
\([\dfrac{du}{dx}][\dfrac{dy}{du}] =\)
\([4][\cos(u)]\)
\(4\cos(4x)\)
