chapter 4 differentiation nhaa/imk/unimap. introduction differentiation – process of finding the...
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DERIVATIVE OF A POWER FUNCTION If n is an integer, then: NHAA/IMK/UNIMAPTRANSCRIPT
NHAA/IMK/UNIMAP
CHAPTER 4
DIFFERENTIATION
NHAA/IMK/UNIMAP
INTRODUCTION • Differentiation
– Process of finding the derivative of a function.
• Notation
yDydxdy
xfxfdxd
x,,
,
NHAA/IMK/UNIMAP
DERIVATIVE OF A POWER FUNCTION • If n is an integer, then:
1 nn nxxdxd
NHAA/IMK/UNIMAP
DERIVATIVE OF A CONSTANT If f is differentiable at function x and c is any real number, then c is differentiable:
xfdxdcxcf
dxd
NHAA/IMK/UNIMAP
Example 1Differentiate the following function:
10
2
3
3
4
2
xxfc
xxfb
xxfa
NHAA/IMK/UNIMAP
DERIVATIVE OF SUM AND DIFFERENCE RULES
If f and g are differentiable at function x, then the function f+g and f-g are differentiable:
xgdxdxf
dxdxgxf
dxd
xgdxdxf
dxdxgxf
dxd
NHAA/IMK/UNIMAP
Example 2Differentiate the following function:
20
3
2
5
23
3
xxfc
xxxfb
xxxfa
NHAA/IMK/UNIMAP
Derivative of Trigonometric Functions
xfy )(xfdxdy
xsin
xcos
kxsin
kxcos
xcos
xsin
kxk cos
kxk sin
NHAA/IMK/UNIMAP
DERIVATIVE OF EXPONENTIAL & LOGARITHMIC FUNCTIONS
xfy )(xfdxdy
xe
xln x1
xeaxe axae
axln x
axdxd
ax11
NHAA/IMK/UNIMAP
PRODUCT RULEIf u and v are differentiable at function x, then so the product u.v, thus
udxdvv
dxduuv
dxd
NHAA/IMK/UNIMAP
Example 3:Differentiate the following function:
xeyd
xeycxxyb
xxxya
x
x
5ln2
4sin32
3
3
2
42
NHAA/IMK/UNIMAP
QUOTIENT RULEIf u and v are differentiable at function x, then is also differentiable
2v
vdxduu
dxdv
vu
dxd
vu
NHAA/IMK/UNIMAP
Example 4Differentiate the following function:
x
eycx
xybxx
xya
x
3sin
2cos
21
4
33
2
NHAA/IMK/UNIMAP
Example 5Differentiate and SIMPLIFY the following function:
xxyd
xeyc
xxyb
xeya
x
x
3cos12sin
sin4
2sin2
cos
3
3
2
NHAA/IMK/UNIMAP
Example 6Differentiate the following function:
xeydx
eyc
exyb
xxya
x
x
x
3ln3sin
2
ln
3
2
33
2
NHAA/IMK/UNIMAP
COMPOSITE FUNCTIONThe Chain Rule
– If g is differentiable at point x and f is differentiable at the point g(x), then is differentiable at x.
– Let and , then xgfy
dxdu
dudy
dxdy
gf xgu
NHAA/IMK/UNIMAP
Example 7Differentiate the following function:
xyb
xya
3sin
13ln
NHAA/IMK/UNIMAP
“Outside-Inside” Rule– Alternative method for Chain Rule:– If ,then xgfy
xgxgfdxdy .
COMPOSITE FUNCTION
NHAA/IMK/UNIMAP
Example 8Differentiate the following function:
x
x
exyc
eyb
xya
22
2sin
ln
3sinln
NHAA/IMK/UNIMAP
• These equation define an implicit relation between variables x and y.
• When we cannot put an equation F(x,y)=0 in the form y = f(x), use implicit differentiation to find
IMPLICIT DIFFERENTIATION
09,025 3322 xyyxyx
dxdy
NHAA/IMK/UNIMAP
• Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x
• Collect the terms with on one side of the equation
• Solve for
IMPLICIT DIFFERENTIATION
dxdy
dxdy