Slide 1 ___________________________________ THE CHAIN RULE
• If f and g are both differentiable and F = f 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)
___________________________________ ___________________________________ “OUTSIDE-INSIDE” RULE
Note: we differentiate the outer function f [at the
inner function g(x)] and then multiply by the
derivative of the inner function.
___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 2 Ex. Differentiate: F(x) =
___________________________________ x2 +1
F(x) = (x2 + 1)1/2
• What is the outer function?
x2 + 1
___________________________________ derivative of
the inside
inside
left alone
F’(x) =
___________________________________ square root function
What is the inside function?
1
−
1
F’(x) = (x2 +1) 2 ( 2x)
●
2
x
___________________________________ ( x 2 + 1)
___________________________________ ___________________________________ ___________________________________ Slide 3 ___________________________________ Ex. Differentiate: y = sin(4x)
• What is the outside function?
sine function
What is the inside function?
y’ = cos(4x)
derivative of outer
function
4
___________________________________ 4x
derivative of inner function
___________________________________ at the inner function
___________________________________ y’ = 4cos(4x)
___________________________________ ___________________________________ ___________________________________ Slide 4 ___________________________________ Ex. Find the derivative.
y = sin(xcosx)
• y’ = cos(xcosx)
•
(cosx – xsinx)
derivative of outside function
inside function left alone
___________________________________ derivative of inside function
(use product rule)
___________________________________ y’ = (cosx – xsinx)[cos(xcosx)]
___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 5 ___________________________________ Example: Find the derivative
y = tan2(3θ)
note: tan2 (3θ) = [tan(3θ)]2
• y’ = 2[tan(3θ)]
•
derivative of outer function
(inner function unchanged )
y’ =
___________________________________ sec2(3θ) • 3
derivative of inner function
___________________________________ 6tan(3θ)sec2(3θ)
___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 6 ___________________________________ Find the derivatives
a) f(x) = sin(x2 + x)
___________________________________ f’(x) = cos(x2 + x)·(2x + 1)
= (2x + 1)cos(x2 + x)
b) h(x) = cos2x
___________________________________ h’(x) = -sin(2x)·2 = -2sin(2x)
___________________________________ c) g(t) = tan(5 – sin2t)
g’(t) = -2cos2t·sec2(5 – sin2t)
___________________________________ ___________________________________ ___________________________________ Slide 7 ___________________________________ Find the derivative:
a)
y'=
y=
sin2 x
cosx
u=sin2 x; u'=2sinxcosx v=cosx; v'=-sinx
cosx ⋅ 2sinxcosx - sin2 x ⋅ ( − sin x)
cos2 x
sinx(2cos2 x + sin2 x)
cos2 x
=
=
=
___________________________________ 2cos2 xsinx + sin3 x
cos2 x
___________________________________ sinx(2cos2 x+1-cos2 x)
cos2 x
sinx(cos2 x+1) = sin x ⎛ cos2 x + 1 ⎞ = sinx 1+sec 2 x
(
)
⎜
2
2 ⎟
⎝ cos x cos x ⎠
cos2 x
___________________________________ ___________________________________ ___________________________________ ___________________________________