Section 2.8: The Derivative as a Function
1. Find the derivative of the function using the definition of derivative (do
not use shortcuts).
(a) \( \displaystyle{f(x)= x^2+2x} \) solution
(b) \( \displaystyle{f(x)= 1- x^2} \) solution
(c) \( \displaystyle{f(x)= x^2-5x+1}\) solution
(d) \( \displaystyle{f(x)= 3x^2-5x }\) solution
(e) \( \displaystyle{f(x)= \frac{1}{x}} \)
solution
(f) \( \displaystyle{f(x)= \frac{1}{\sqrt{x}}} \)
solution
2. Use the definition of a derivative to find \(f'\) and \(f''.\) Given that \(f(x)=3x^2-5x+7.\) solution
3. The graph of \( y= f(x) \) is given below. State, with reasons, the numbers at which \( f \) is not differentiable.
4. Find an equation of the tangent line to the curve \( f(x) =\sqrt{x}
\) at \(x=4\). Given that \( f'(x)=\dfrac{1}{2\sqrt{x}} \).
solution
5. Find an equation of the tangent line to the curve \( f(x)
=2x^2-5x \) at \( (1, -3)\). Given that \( f'(x)= 4x-5 \).
solution
6. The displacement (in meters) of a particle moving in a straight line is given by \( s(t) =t^3-2t\), where \(t\) is measured in seconds.
Given that \(s'(t)=3t^2-2\), and \(s''(t)=6t\).
solution
(a) Find the average velocity over the time interval \( [2, 5]\).
(b) Find the instantaneous velocity at \(t=2\).
(c) Find the acceleration when \( t = 3\).