Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
In my class of Calculus-I, I take lecture note from these slides, hope these lecture slides help other student.The key point in these slides are:Chain Rule, Composite Function, Leibniz Notation, Differentiable Functions, Outer Function, Derivative of Function, Power Rule, Real Number, Implicit Differentiation, Variable Explicitly, Equation of Tangent Line
Typology: Slides
1 / 10
This page cannot be seen from the preview
Don't miss anything!
F ′( x) = f ′(g(x))⋅ g′(x )
dx
du
du
dy
dx
Example:
( (^ )) ( )
2 f g x = sin x − 4
( )
2 y = sin x − 4
y = sinu
2 u = x − 4
cos
dy u du
= 2
du x dx
=
dy dy du
dx du dx
= ⋅
cos 2
dy u x dx
= ⋅
( )
2 cos 4 2
dy x x dx
= − ⋅
( )
2 y = sin x − 4
( ) ( )
2 2 cos 4 4
d y x x dx
′ = − ⋅ −
( )
2 y ′ = cos x − 4 ⋅ 2 x
A faster way to write the solution:
Differentiate the outer function...
…then the inner function
Another example:
( )
2 cos 3
d x dx
( )
2 cos 3
d x dx
2 cos 3 ( ) cos 3( )
d x x dx
2 cos 3 ( ) sin 3( ) ( 3 )
d x x x dx
−2 cos 3 (^) ( x (^) ) ⋅ sin 3( x)⋅ 3
−6 cos 3 ( x (^) ) sin 3( x)
The chain rule can be used
more than once.
(That’s what makes the
“chain” in the “chain rule”!)
−
1 g x n g x g x dx
d (^) n n = ⋅ ′
−
If n is any real number and u=g(x) is differentiable,
then
Alternatively,
Example:
2 50 2 49
2 50 2 49 2
50 3 2 100 3
3 50 3 3
= − ⋅ = −
− = − −
x x x x
x dx
d x x dx
d
2 2 x + y = 1
This is not a function,
but it would still be
nice to be able to find
the slope.
2 2 1
d d d x y dx dx dx
2 2 0
dy x y dx
Note use of chain rule.
2 2
dy y x dx
= −
2
2
dy x
dx y
dy x
dx y
= −
Example 1:
2 2 y = x +sin y
2 2 sin
d d d y x y dx dx dx
= +
This can’t be solved for y.
2 2 cos
dy dy x y dx dx
= +
2 cos 2
dy dy y x dx dx
− =
( 2 cos ) 2
dy y x dx
− =
2
2 cos
dy x
dx y
= −
Example 2:
Higher Order Derivatives
Find if.
2
2
d y
dx
3 2 2 x − 3 y = 7
3 2 2 x − 3 y = 7
2 6 x − 6 y y′ = 0
2 − 6 y y ′ = − 6 x
2 6
x y y
2 x y y
2
2
y 2 x x y y y
2
2
2 x x y y y y
2 2
2
2 x x y y
x
y y
4
3
2 x x y y y
Substitute
back into the
equation.
y′
