Derivace-příklady
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v′ =
ln2 x.
c
Robert Maˇr´ık, 2004.
Differentiate y = x ln2 x.
y′ = ( x
ln2 x )′ = (x)′ ln2 x + x (ln2 x)′
= 1 ln2 x + x 2 ln x (ln x)′
= ln2 x + x 2 ln x
1
x
= (2 + ln x) ln x
Derivative of x is formula.
c
Robert Maˇr´ık, 2004.
Differentiate y = x ln2 x.
y′ = ( x
ln2 x )′ = (x)′ ln2 x + x (ln2 x)′
= 1 ln2 x + x 2 ln x (ln x)′
= ln2 x + x 2 ln x
1
x
= (2 + ln x) ln x
The function ln2 x is a composite function (ln x)2. The
outside function is the quadratic function and the inside
function is the logarithmic function. We use the chain
rule.
c
Robert Maˇr´ık, 2004.
Differentiate y = x ln2 x.
y′ = ( x
ln2 x )′ = (x)′ ln2 x + x (ln2 x)′
= 1 ln2 x + x 2 ln x (ln x)′
= ln2 x + x 2 ln x
1
x
= (2 + ln x) ln x
The derivative of logarithm is a formula.
c
Robert Maˇr´ık, 2004.
Differentiate y = x ln2 x.
y′ = ( x
ln2 x )′ = (x)′ ln2 x + x (ln2 x)′
= 1 ln2 x + x 2 ln x (ln x)′
= ln2 x + x 2 ln x
1
x
= (2 + ln x) ln x
x
1
x
= 1 and the common factor ln x can be taken out.
c
Robert Maˇr´ık, 2004.
Differentiate y = x ln2 x.
y′ = ( x
ln2 x )′ = (x)′ ln2 x + x (ln2 x)′
= 1 ln2 x + x 2 ln x (ln x)′
= ln2 x + x 2 ln x
1
x
= (2 + ln x) ln x
Finished!
Problem 5, y =
3
s
1 + x3
1 − x3
c
Robert Maˇr´ık, 2004.
Differentiate y =
3
s
1 + x3
1 − x3
.
y′ =
1
3
1 + x3
1 − x3
−2/3 1 + x3
1 − x3
′
=
1
3
1 − x3
1 + x3
2/3
(1 + x3)′(1 − x3) − (1 + x3)(1 − x3)′
(1 − x3)2
=
1
3
1 − x3