10. Let f(x) = sin r. Use the definition of the derivative to prove that f'(x) = cos r.

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11. Let f(1) = cos r. Use the definition of the derivative to prove that f'(:1) = - sin r.

11. Let f(1) = cos 1. Use the definition of the derivative to prove that f'(:1) = -sin . Solution: We have · f'(:1) = lim cos(2+h) - cos(2) +0 cos(1) cos(h) - sin() sin(h) - cos(1) = lim h0 sin / – Jin contar) (cos(1) = -1) = sin(e) (sinm) h0 sin(h) cos(h) - = lim cos(2) h cos(h) - 1 = cos(x) lim h- oh = cos(x)0 - sin(x)(1) = - sin h c) lim 0 / h

f(1) [3] 8. Let f(x) = 2x2 +3.r. Use the definition of the derivative to prove that f'(x) = 4x+3. Solution: We have f(x) - f(a) 2,c2 + 3.r - 2a? - 3a 1+ I -a 1 r - a - lim 2(x + a)(x - a) + 3(x -a) -a = lim 2(+ a) +3 = 4a +3 = lim a 1-a - as required.

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10. Let f(x) = sin r. Use the definition of the derivative to prove that f'(x) = cos r.

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