Derivatives Practice Worksheet 2 - Math 1A, section 103
1. Find the slope of the tangent line at the point (2, 4) on the curve y = x2.
2. Let f(x) = x3 - 2x.
(a) Draw the graph of f(x). Where, precisely, does it cross the x-axis?
(b) Using your graph from part (a), sketch the graph of the derivative f'(x).
(c) Now take the derivative of f using derivative rules, and draw the resulting graph. Does it match your picture from part (b)?
3. Compute the derivative of the following functions with respect to x.
(a) f(x) = sin(x)/1+cos(x)
(b) f(x) = tan(x)2
(c) f(x) = 6x3+3x-1/x3
(d) f(x) = sin(x)/x
(e) f(x) = sin(x) cos(y)
4. Chain rule: The chain rule says that the derivative of the composition of two functions, f(g(x))', is equal to g'(x) · f'(g(x)). Use the chain rule to compute the derivatives of the following functions:
(a) f(x) = sin(2x)
(b) f(x) = tan(x)2
(c) f(x) = e3x^2
(d) f(x) = 1/(1 + x)
(e) f(x) = sin(cos(x))
5. Derive the "triple chain rule": what is the derivative of f(g(h(x)))?
6. An iPhone is thrown into the air, starting from a height of 1 meter off the ground, and with an initial velocity of 2 meters per second (written 2m/s). Gravity slows it down by an acceleration of 9.8 meters per second squared (9.8m/s2).
(a) What is the maximum height reached by the iPhone, and how long does it take the iPhone to reach that height? (Hint: The object's height as a function of time is given by h(t) = h0 + vt - 1/2at2 where h0 is the initial height, v is the initial velocity, and a is the velocity.)
(b) If you don't catch the iPhone on the way down, how long does it take for it to smash against the ground?