Derivative Substitution - # 2
Posted: Mon Apr 27, 2026 10:29 am
\(In\,\,The\,\,Case\,\,Of\,\,Nested\,\,Functions:\)
\(\dfrac{dy}{dx} = [\dfrac{du}{dx}][ \dfrac{dy}{du}]\)
\(ex. a\)
\(\dfrac{d}{dx}[\cos^{2}(2x)]\)
\(u = \cos(2x)\)
\(\dfrac{dy}{dx} = [\dfrac{dy}{du}][\dfrac{du}{dx}]\)
\(\dfrac{dy}{dx} = [2(u)][-2\sin(2x)]\)
\(\dfrac{dy}{dx} = 2(\cos(2x))(-2\sin(2x))\)
\(ex. b\)
\(\dfrac{d}{dx}[\ln(5x)]\)
\(u = 5x\)
\(\dfrac{dy}{dx} = [\dfrac{dy}{du}][\dfrac{du}{dx}]\)
\(\dfrac{dy}{dx} = [\dfrac{1}{u}][5]\)
\(\dfrac{dy}{dx} = 5\dfrac{1}{5x} = \dfrac{5}{5x} = \dfrac{1}{x}\)
\(\dfrac{dy}{dx} = [\dfrac{du}{dx}][ \dfrac{dy}{du}]\)
\(ex. a\)
\(\dfrac{d}{dx}[\cos^{2}(2x)]\)
\(u = \cos(2x)\)
\(\dfrac{dy}{dx} = [\dfrac{dy}{du}][\dfrac{du}{dx}]\)
\(\dfrac{dy}{dx} = [2(u)][-2\sin(2x)]\)
\(\dfrac{dy}{dx} = 2(\cos(2x))(-2\sin(2x))\)
\(ex. b\)
\(\dfrac{d}{dx}[\ln(5x)]\)
\(u = 5x\)
\(\dfrac{dy}{dx} = [\dfrac{dy}{du}][\dfrac{du}{dx}]\)
\(\dfrac{dy}{dx} = [\dfrac{1}{u}][5]\)
\(\dfrac{dy}{dx} = 5\dfrac{1}{5x} = \dfrac{5}{5x} = \dfrac{1}{x}\)