

Subscribe to our Youtube Channel - https://you.tube/teachoo
Last updated at Dec. 7, 2020 by Teachoo
Transcript
Example 20 Find the derivative of f(x) from the first principle, where f(x) is (i) sin x + cos x Given f (x) = sin x + cos x We need to find Derivative of f(x) We know that fβ(x) = limβ¬(hβ0) πβ‘γ(π₯ + β) β π(π₯)γ/β Here, f (x) = sin x + cos x f (x + h) = sin (x + h) + cos (x + h) Putting values fβ(x) = limβ¬(hβ0)β‘γ(sinβ‘γ(π₯ + β)γ + cosβ‘(π₯ + β) β (sinβ‘π₯ + cosβ‘γπ₯)γ)/βγ Using sin (A + B) = sin A cos B + cos A sin B & cos (A + B) = cos A cos B β sin A sin B = limβ¬(hβ0)β‘γsinβ‘γπ₯ cosβ‘γβ +γ cosγβ‘γπ₯ sinβ‘γβ + cosβ‘γπ₯ cosβ‘γβ β sinβ‘γπ₯ γ sinγβ‘γβ βγ sinγβ‘γπ₯ βγ cosγβ‘π₯ γ γ γ γ γ γ γ γ γ/hγ = limβ¬(hβ0)β‘γcosβ‘γπ₯ sinβ‘γβ βγ sinγβ‘γπ₯ sinβ‘γβ + sinβ‘γπ₯ cosβ‘γβ β sinβ‘γπ₯ +γ cosγβ‘γπ₯ cosβ‘γβ βγ cosγβ‘π₯ γ γ γ γ γ γ γ γ γ/hγ = limβ¬(hβ0)β‘γsinβ‘γβ γ(cosγβ‘γπ₯ β sinβ‘γπ₯) + sinβ‘γπ₯ (cosβ‘γβ β 1) + cosβ‘π₯ (cosβ‘γβ β 1)γ γ γ γ γ γ/hγ = limβ¬(hβ0)β‘(sinβ‘γβ γ(cosγβ‘γπ₯ β sinβ‘γπ₯)γ γ γ/h+sinβ‘γπ₯ (cosβ‘γβ β 1)γ γ/h+cosβ‘γπ₯ (cosβ‘γβ β 1)γ γ/β)" " = limβ¬(hβ0)β‘γsinβ‘γβ γ(cosγβ‘γπ₯ βγ sinγβ‘γπ₯)γ γ γ/h+limβ¬(hβ0) sinβ‘γπ₯ (cosβ‘γβ β 1)γ γ/h+limβ¬(hβ0) cosβ‘γπ₯ (cosβ‘γβ β 1)γ γ/βγ = limβ¬(hβ0)β‘γ"(cos x β sin x)" sinβ‘β/β+limβ¬(hβ0) "(β sin x)" ((1 β cosβ‘γβ)γ)/βγ+limβ¬(hβ0) "(β cos x)" ((1 β cosβ‘γβ)γ)/β = "(cos x β sin x)" (π₯π’π¦)β¬(π‘βπ)β‘γπ¬π’π§β‘π/πβsinβ‘γπ₯ (π₯π’π¦)β¬(π‘βπ) γ ((π β πππβ‘γπ)γ)/πβcosβ‘π₯ (π₯π’π¦)β¬(π‘βπ) ((π β πππβ‘γπ)γ)/πγUsing (πππ)β¬(ββ0) π ππβ‘β/β = 1 & (πππ)β¬(ββ0) γ(1 β πππ γβ‘γβ)γ/β = 0 Using (πππ)β¬(ββ0) π ππβ‘β/β = 1 & (πππ)β¬(ββ0) γ(1 β πππ γβ‘γβ)γ/β = 0
Examples
Example 2 Important
Example 3 Important
Example 4
Example 5
Example 6
Example 7 Important
Example 8
Example 9
Example 10
Example 11
Example 12
Example 13
Example 14
Example 15
Example 16
Example 17 Important
Example 18
Example 19 Important
Example 20 (i) Important You are here
Example 20 (ii)
Example 21 Important
Example 22 Important
About the Author