Chapter 3 Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
1. Derivative of a constant:
Differentiate with respect to x. Watch for constants!
2. The Power Rule:
Power rule: If is a rational number and, then
Find the first derivative of each function with respect to the given variable.
3. Differentiate with respect to:
4. Find the equation of the tangent and normal (perpendicular) line to each curve at the given point. or x-value.
5. Find the x- coordinate(s) for where the function attains its local maxima and minima if any (Hint: Find the derivative of the function and set it equal to zero). (Local max and min are also known as “turning points”). Also implies that the function has horizontal tangents at this x values.
6. Find the first and second derivatives for each function.
7. for what value(s) of, is the instantaneous rate of change (velocity = derivative) of equal to the instantaneous rate of change of
8. Find the second derivative of each function.
(a) (b) (c)
9. If ,then determine the value of, such that is both, differentiable and continuous (Hint: Set equations for f(x) shown, as well as for the first derivative, equal to each other.).
10. Find the average rate (average velocity) of each function on the given interval and the instantaneous rate when .
11. Evaluate the following derivatives for the given x value. What does the value of the derivative represent (Hint: Slope)?
12. Find the equation of the normal line to when x = 1.
13. Given the curve C with equation ,
(a)Show that the rate of change (slope) of the function with respect to is
(b)The point H on C has x – coordinate 2, and the tangent line to C at H intercepts the x –axis at the point (k, 0). Find the value of k.
14 Given that and that find the equation of the line tangent to the graph of
15. Given . Find
(a) The velocity (1st derivative) and the acceleration (2nd derivative) of the function.
(b) The velocity after 3 seconds.
(c) The acceleration is zero.
(d) The maximum speed of the particle on the interval [0, 5].
3.2 The Product and Quotient Rules
Write the shortcut Product rule using u and v.
Write the shortcut Quotient rule using u and v.
2. If , find
3. Find the value of k, where and
3.3 Derivatives of Trigonometric Functions
by the Squeeze Theorem.
1. Evaluate the following limits:
(a) =(b)(c) (d)
Derivatives of trigonometric functions:
2. Find the following derivatives with respect to Rewrite when necessary.
(a) (b) (c)
3. Find the tangent and normal line to the derivative of at the indicated point.
(d) (e) (f)
5. Find if, and where
(a) (b) (c)
3.4 The Chain Rule
(a) (b) (c)
2. Find the rate of change of the function when .
3. Differentiate with respect to x:
5. A particle with position , starts at t = 0 and stops at t = 10. Use a calculator.
(a) When is the particle moving to the left? Right?
(b) When does the particle reach its maximum speed?
(c) For what value of t is the particle at rest?
(d) When is the speed of the particle increasing and decreasing?
6. ? when ,
7. Use the calculator to evaluate the following derivatives. Compute the values graphically and also by using the NDeriv ( ) command.
(a) , (b)
3.5 Implicit Differentiation
Explicit functions:Implicit functions:
Some implicit functions can be written as explicit functions. For example:
1. In examples (a) – (d) differentiate with respect to, meaning.
2. Find for
3. Find the y – intercept of the tangent line to the hyperbola at the point
4. Determine the points where has horizontal and vertical tangent line(s).
5. Determine the points where the derivative of is undefined.
6. If then for is
7. Make sure each answer does not contain derivatives.
(a) If , find .(b).
8. Differentiate with respect to x.
9. Find the exact value of each expression.
(a) (b) (c) (d) (e) (f)
10. Find (well….look these up) the derivatives of the six inverse trigonometric functions.
(a) =(b) =(d) =
(d) (e) =(f)
(a) (c) (d) (e) (h) (j)
12. If is a solution to the differential equation find .
3.6 Logarithmic Differentiation
1. Differentiate with respect to x.
(a) (b) (c) (d)
(a) (b) (c)
4. Find for (a)(b)
5. Find each derivative using logarithmic differentiation.
(a) (b) (c)
6. Consider the function .
(a) Find the average rate of change of the function on the interval
(b) For what value(s) of is the average rate of change of change of equal the instantaneous rate of change?
7. A curve H whose equations are given by .
(a) Show that
(b) Find the equation of the tangent line to when
8. Using the alternate form of the derivative, find
9. Use the alternate form of the derivative to find for each function:
10. Identify the intervals where the “derivative” is increasing and decreasing. Approximate the points if any, where the derivative changes from increasing to decreasing.
(a) (b) (c)
3.7 Rates of Change
1. Sketch the velocity and the acceleration of teach graph given its position function.
In part (a) the path is described by the equation, for . Find the interval where the particle’s speed is increasing and decreasing.
2. The velocity of a particle is given below. Is the speed of each particle increasing? Decreasing? Write an explanation.
(a) (b) (c)
3. The graph of is graphed, what can you say about
4. Sketch the graph of the derivative of each function. Approximate the points at which the velocity changes from increasing to decreasing.
Use the following graph to determine the approximate points where graph changes concavity. Use this to determine on what intervals the speed is increasing and decreasing.
3.8 Exponential growth and decay
Let be the value of a quantity y at time t and if the rate of change of with respect to is proportional to its size at any time, then where is a constant. The differential equation is known as the law of natural growth and decay.
Where A represents the initial amount. Note that at t = 0, y(0) = A
THE FOLLOWING QUESTIONS WE WILL DO TOGETHER IN CLASS
1. A bacteria culture starts with 500 bacteria and grows at a rate proportional t its size. After 3 hours there are 8000 bacteria.
(a) Find an expression for the amount of bacteria after 4 hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 30,000?
2. Bismuth- 210 has a half-life of 5 days.
(a) A sample originally has a mass of 800 mg. Find a formula for the mass remaining after t days.
(b) Find the mass remaining after 30 days.
(c) When is the mass reduced to 1 mg?
(d) Sketch the graph of the mass function.
Newton’s Law of Cooling
The rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. ====> Temp of the environment
3. A bottle of soda pop at room temperature () is placed in a refrigerator where the temperature is 44 degrees Fahrenheit. After ½ an hour the soda pop has cooled to.
(a) What is the temperature of the soda pop after another half an hour?
(b) How long does it take for the soda pop to cool down to?
4. A roast turkey is taken from an oven when its temperature has reached 185 degrees Fahrenheit and is placed on a table in a room where the temperature is 75 degrees Fahrenheit.
(a) If the temperature of the turkey is 150 degrees Fahrenheit after half and hour, what is the temperature after 45 min?
(b) When will the turkey have cooled to 100 degrees Fahrenheit?
3.9 Related Rates
Assume the variables given are differentiable functions of , unless you are told that they are constant.
2. Example: ==>
5. Air is pumped into a spherical balloon at a rate of . What is the rate of change of the radius at the instant when the radius is 3 cm?
6. A ladder 13 ft long is sliding down a wall at a rate of 3 feet per second. At what rate is the ladder moving away from the wall, when the ladder is 5 ft from the wall?
7. If and find when and y > 0.
8. A person is at point A, which is 40 feet from point C. A balloon rises from point C at 4 feet per second and its height is the point B.
(a)What is the rate of change of the distance between the person and the balloon at the time that it has risen 30 feet?
(b)What is the rate of change of the area of triangle ABC at feet?
(c)What is the rate of change of the angle of elevation, theta, when the balloon has risen 30 feet? See picture.
9. If find the value of
10. Find at the point where for .
11. Water is running out tank in the shape of a cone. The diameter of the tank is 8 feet and the altitude is 16 feet, find the rate at which the water level is dropping when the water is 4 feet from the top and the water is running out at a rate of 3 cubic feet per second.
12. A man 6 ft tall, is walking toward a street light 30 feet high, at a rate of 3 feet per second.
(a) At what rate is the tip of his shadow changing?
(b) At what rate is the length of his shadow changing?
3.10 Linear Approximations
Find the linearization L(x) of each function at Means: find the tangent line at
1. Find the tangent line to and use it the tangent line to approximate the value of .
2. Find the linearization of and use the tangent line to approximate the value of .
3. Find the differentials.
(a) (b) (c)
4. Identify the function and where the derivative of this function is being evaluated.