Derivative as a Function

1. Find the derivative using the limit definition:

1(a). \( \displaystyle{f(x)= x^2+2x} \)
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1(b). \( \displaystyle{f(x)= 1- x^2} \)
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1(c). \( \displaystyle{f(x)= x^2-5x+1}\)
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1(d). \( \displaystyle{f(x)= 3x^2-5x }\)
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1(e). \( \displaystyle{f(x)= \frac{1}{x}} \)
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1(f). \( \displaystyle{f(x)= \frac{1}{\sqrt{x}}} \)
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2. Use the definition of a derivative to find \(f'\) and \(f''.\) Given that \(f(x)=3x^2-5x+7.\)
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3. The graph of \( y = f(x) \) is given below. State, with reasons, the numbers at which \( f \) is not differentiable. Graph of a function with corners, discontinuities, or vertical tangents.

Graph of a function with corners, discontinuities, or vertical tangents.

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4. Find an equation of the tangent line to the curve \( f(x) =\sqrt{x} \) at \(x=4\). Given that \( f'(x)=\dfrac{1}{2\sqrt{x}} \).
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5. Find an equation of the tangent line to the curve \( f(x) = 2x^2-5x \) at \( (1, -3)\). Given that \( f'(x)= 4x-5 \).
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6. A particle moves in a straight line with displacement \( s(t) =t^3-2t\). Given \(s'(t)=3t^2-2\) and \(s''(t)=6t\):

(a) Find the average velocity over the time interval \( [2, 5]\).
(b) Find the instantaneous velocity at \(t=2\).
(c) Find the acceleration when \( t = 3\).

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