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Last updated at March 22, 2023 by Teachoo

Example 7 Find the derivative of sin x at x = 0. Let f(x) = sin x We know that fβ(x) = (πππ)β¬(ββ0) πβ‘γ(π₯ + β) β π(π₯)γ/β Here, f(x) = sin x f(x + h) = sin (x + h) Now, fβ(x) = limβ¬(hβ0) π ππβ‘γ(π₯ + β) β π ππ π₯γ/β Putting x = 0 fβ (0) = limβ¬(hβ0) π ππβ‘γ(0 + β) β π ππ (0)γ/β = limβ¬(hβ0) sinβ‘γβ β 0γ/h = limβ¬(hβ0) sinβ‘γβ γ/h = 1 Hence, derivative of sin x at x = 0 is 1 Using limβ¬(xβ0) sinβ‘π₯/π₯ = 1 Replacing x by h limβ¬(xβ0) sinβ‘β/β = 1