3.1.1-eg - Example Let f(x = 3x Use the denition of the...

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Example Let f ( x ) = 3 x . Use the definition of the derivative to find f 0 (3) = lim h 0 f (3 + h ) - f (3) h , (1) and then write the equation of the tangent line to the curve y = f ( x ) at the point (3 , f (3)). Solution: Making the substitution f (3 + h ) = q 3(3 + h ) into (1), it follows that f 0 (3) = lim h 0 q 3(3 + h ) - 3 · 3 h = lim h 0 9 + 3 h - 3 h . Multiplying numerator and denominator in the above by the conjugate expression, 9 + 3 h + 3, to 9 + 3 h - 3, we have f 0 (3) = lim h 0 9 + 3 h - 3 h · 9 + 3 h + 3 9 + 3 h + 3 = lim h 0 (9 + 3 h ) - 9 h 9 + 3 h + 3 = lim h 0 3 h h 9 + 3 h + 3 = lim h 0 3
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