CHAPTER Continuity The Chain Rule

CHAPTER Continuity The Chain Rule If f and g are both differentiable and F = f o g is the composite function defined by F(x) =f(g(x)), then F is differentiable and F’ is given by the product F(x) = f’(g(x)) g’(x). In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then dy / dx = ( dy/du ) ( du/dx ).

CHAPTER Continuity Example Write the composite function in the form f (g(x)). Then find dy /dx. a) y = cos 2 x b) y = sin (sin x).

CHAPTER Continuity The Power Rule Combined with the Chain Rule If n is any real number and u = g(x) is differentiable, then d / dx ( u n ) = n u n-1 ( du/dx ). Alternatively, d / dx [ g(x) ] n = n [ g(x) ] n-1 g’(x).

CHAPTER Continuity Example Differentiate f (t) = 1 / ( t 2 – 2t – 5 ) 4.

CHAPTER Continuity d / dx ( a x ) = a x ln a Example Differentiate y = 3 x. Example Differentiate y = 2 3 x.

CHAPTER Continuity Tangents to Parametric Curves dy / dx = ( dy / dt ) / ( dx / dt ) if dx / dt is not equal to 0.

CHAPTER Continuity Example The cycloid has the parametric equations x=r(  - sin  ), y=r(  - cos  ). Find the equation of the tangent to the cycloid at the point where  =  / 3.

CHAPTER Continuity Example On what interval is the curve y = e -x 2 concave downward?