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**
Question 16 (OR 1
**
**
st
**
**
question)
**

If y = x
^{
sin β‘x
}
+ sinβ‘ (x
^{
x
}
), find dy/dx

Last updated at Oct. 1, 2019 by Teachoo

Subscribe to our Youtube Channel - https://www.youtube.com/channel/UCZBx269Tl5Os5NHlSbVX4Kg

**
Question 16 (OR 1
**
**
st
**
**
question)
**

If y = x
^{
sin β‘x
}
+ sinβ‘ (x
^{
x
}
), find dy/dx

Transcript

Question 16 (OR 1st question) If y = π₯^sinβ‘π₯ +sinβ‘γ(π₯^π₯)γ, find ππ¦/ππ₯ Let u = π₯^sinβ‘π₯ , π£=sinβ‘γ(π₯^π₯)γ Thus, y = u + v Differentiating π€.π.π‘.π₯ ππ¦/ππ₯ = ππ’/ππ₯ + ππ£/ππ₯ Calculating derivative of u and v separately Solving π π/π π u = π₯^sinβ‘π₯ Taking log both sides logβ‘π’ = log π₯^sinβ‘π₯ logβ‘π’ = sinβ‘π₯ . log π₯ Differentiating π€.π.π‘.π₯ (π(logβ‘γπ’)γ)/ππ₯ = π/ππ₯ (sinβ‘γπ₯ logβ‘π₯ γ ) By product Rule (uv)β = uβv + vβu where u = sin x & v = log x (π(logβ‘γπ’)γ)/ππ₯ = (π(sinβ‘π₯))/ππ₯.log π₯+sin π₯ . (π(logβ‘π₯))/ππ₯ (π(logβ‘γπ’)γ)/ππ’ Γ ππ’/ππ₯ = cosβ‘π₯ logβ‘π₯ + sinβ‘π₯ 1/π₯ 1/π’ " Γ " ππ’/ππ₯ = cos π₯β‘logβ‘π₯ + sinβ‘π₯ 1/π₯ ππ’/ππ₯ = π’ (γcos xγβ‘γlogβ‘γπ₯+ 1/π₯γ sinβ‘π₯ γ ) Putting back π’ = π₯^π ππβ‘π₯ ππ’/ππ₯ = π₯^π ππβ‘π₯ (cosβ‘γlogβ‘γπ₯+ 1/π₯γ sinβ‘π₯ γ ) Solving π π/π π v = sinβ‘γ(π₯^π₯)γ Let t = π₯^π₯ Taking log both sides logβ‘π‘ = log π₯^π₯ logβ‘π‘ = π₯ log π₯ Differentiating π€.π.π‘.π₯ (π(logβ‘γπ‘)γ)/ππ₯ = π/ππ₯ (π₯ logβ‘π₯ ) (π(logβ‘γπ‘)γ)/ππ₯ = (π(π₯))/ππ₯.log π₯+π₯ . (π(logβ‘π₯))/ππ₯ (π(logβ‘γπ‘)γ)/ππ₯ = (π(π₯))/ππ₯.log π₯+π₯ . (π(logβ‘π₯))/ππ₯ (π(logβ‘γπ‘)γ)/ππ‘ Γ ππ‘/ππ₯ = 1.log π₯+π₯Γ1/π₯ 1/t Γ ππ‘/ππ₯ = log π₯+1 ππ‘/ππ₯ = t(log π₯+1) Putting t = xx ππ‘/ππ₯ = π₯^π₯ (πππ π₯+1) Now, v = sinβ‘γ(π₯^π₯)γ v = sinβ‘γ(π‘)γ Differentiating w.r.t. x ππ£/ππ₯ = (π(sinβ‘π‘))/ππ₯ ππ£/ππ₯ = (π(sinβ‘π‘))/ππ‘ Γ ππ‘/ππ₯ ππ£/ππ₯ = cos t Γ ππ‘/ππ₯ ππ£/ππ₯ = cosβ‘γ(π₯^π₯)γ (πππ π₯+1)π₯^π₯ Now, from (1) ππ¦/ππ₯ = ππ’/ππ₯ + ππ£/ππ₯ ππ¦/ππ₯ = π₯^π ππβ‘π₯ (cosβ‘γlogβ‘γπ₯+ 1/π₯γ sinβ‘π₯ γ )+ cosβ‘γγ(π₯γ^π₯)γ (πππ π₯+1)π₯^π₯

CBSE Class 12 Sample Paper for 2019 Boards

Paper Summary

Question 1

Question 2

Question 3

Question 4 (Or 1st)

Question 4 (Or 2nd)

Question 5

Question 6

Question 7

Question 8 (Or 1st)

Question 8 (Or 2nd)

Question 9

Question 10 (Or 1st)

Question 10 (Or 2nd)

Question 11

Question 12 (Or 1st)

Question 12 (Or 2nd)

Question 13 (Or 1st)

Question 13 (Or 2nd)

Question 14

Question 15

Question 16 (Or 1st) You are here

Question 16 (Or 2nd)

Question 17

Question 18

Question 19

Question 20

Question 21 (Or 1st)

Question 21 (Or 2nd)

Question 22

Question 23

Question 24 (Or 1st)

Question 24 (Or 2nd)

Question 25

Question 26 (Or 1st)

Question 26 (Or 2nd)

Question 27 (Or 1st)

Question 27 (Or 2nd)

Question 28

Question 29

Class 12

Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.