using properties of log in differentiation

1. LOGARITHMIC
DIFFERENTIATION

2. Question 1
𝑖𝑓 𝑒 𝑥+𝑦
= 𝑥𝑦,
Show that
𝑑𝑦
𝑑𝑥
=
𝑦(1−𝑥)
𝑥(𝑦−1)
𝑒 𝑥+𝑦
= 𝑥𝑦
𝑥 + 𝑦 = log 𝑥 + log 𝑦
Differentiate both sides w.r.t x

3. 1 +
𝑑𝑦
𝑑𝑥
=
1
𝑥
+
𝑑𝑦
𝑑𝑥
1
𝑦
𝑑𝑦
𝑑𝑥
1 −
1
𝑦
=
1
𝑥
− 1
𝑑𝑦
𝑑𝑥
𝑦 − 1
𝑦
=
1 − 𝑥
𝑥
𝑑𝑦
𝑑𝑥
=
𝑦(1 − 𝑥)
𝑥(𝑦 − 1)

4. Question 2
𝑖𝑓 𝑦 = 𝑥 𝑦
, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑥
𝑑𝑦
𝑑𝑥
=
𝑦2
1 − 𝑦𝑙𝑜𝑔𝑥
log y = y log x
1
𝑦
𝑑𝑦
𝑑𝑥
=
𝑦
𝑥
+ 𝑙𝑜𝑔𝑥
𝑑𝑦
𝑑𝑥

5. 𝑑𝑦
𝑑𝑥
(
1
𝑦
− log 𝑥 ) =
𝑦
𝑥
𝑑𝑦
𝑑𝑥
1 − 𝑦𝑙𝑜𝑔𝑥
𝑦
=
𝑦
𝑥
𝑑𝑦
𝑑𝑥
=
𝑦2
𝑥(1 − 𝑦 log 𝑥)
𝑥
𝑑𝑦
𝑑𝑥
=
𝑦2
𝑥(1 − 𝑦𝑙𝑜𝑔𝑥)

6. Question 3
Differentiate 𝑥 𝑠𝑖𝑛𝑥
, 𝑥 > 0 𝑤. 𝑟. 𝑡 𝑥
y = 𝑥 𝑠𝑖𝑛𝑥
log 𝑦 = 𝑠𝑖𝑛𝑥 log 𝑥
1
𝑦
𝑑𝑦
𝑑𝑥
=
𝑠𝑖𝑛𝑥
𝑥
+ 𝑙𝑜𝑔𝑥 𝑐𝑜𝑠𝑥

7. 𝑑𝑦
𝑑𝑥
= 𝑦(
𝑠𝑖𝑛𝑥
𝑥
+ 𝑙𝑜𝑔𝑥 𝑐𝑜𝑠𝑥)
𝑑𝑦
𝑑𝑥
= 𝑥 𝑠𝑖𝑛𝑥(
𝑠𝑖𝑛𝑥
𝑥
+ 𝑙𝑜𝑔𝑥 𝑐𝑜𝑠𝑥)
Question 4
𝑥 𝑦
= 𝑒 𝑥−𝑦
. Prove that
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔𝑥
(𝑙𝑜𝑔𝑥𝑒)2

8. 𝑦 𝑙𝑜𝑔𝑥 = 𝑥 − 𝑦
𝑦
𝑥
+ 𝑙𝑜𝑔𝑥
𝑑𝑦
𝑑𝑥
= 1 −
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
𝑙𝑜𝑔𝑥 + 1 = 1 −
𝑦
𝑥
𝑑𝑦
𝑑𝑥
𝑙𝑜𝑔𝑥 + 𝑙𝑜𝑔𝑒 =
𝑥 − 𝑦
𝑥

9. 𝑑𝑦
𝑑𝑥
=
𝑥 − 𝑦
𝑥 log(𝑥𝑒)
𝑑𝑦
𝑑𝑥
=
𝑦𝑙𝑜𝑔𝑥
𝑥 log(𝑥𝑒)
𝑛𝑜𝑤 𝑦𝑙𝑜𝑔𝑥 = 𝑥 − 𝑦
𝑦 𝑙𝑜𝑔𝑥 + 𝑦 = 𝑥
-------------1

10. 𝑦 𝑙𝑜𝑔𝑥 + 1 = 𝑥
𝑦 log 𝑥𝑒 = 𝑥
𝑦
𝑥
=
1
log(𝑥𝑒)
Substituting in 1,
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔𝑥
(𝑙𝑜𝑔𝑥𝑒)2
**********************

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