3. Techniques of Differentiation

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(s) = 2√s-1; a=25

Use the graph of f(x)=|x| to find f′(x).

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 4x²+1; a= 2,4

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 1/√t; a=9, 1/4

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(s) = 4s³+3s; a= -3, -1

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 3t⁴; a= -2, 2

Find and simplify the derivative of the following functions.

f(x) = 3x^-9

Let F(x) = f(x) + g(x),G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x), where the graphs of f and g are shown in the figure. Find each of the following.

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H'(2)

Use the table to find the following derivatives.

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d/dx (f(x) + g(x)) ∣x=1

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives. <IMAGE>

F'(2)

{Use of Tech} Equations of tangent lines

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = −3x²+2; a=1

{Use of Tech} Equations of tangent lines

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = e^x; a = ln 3

Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W = 1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh = 3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t + 4t² − (t³ / 9) kWh where t = 0 corresponds to midnight.

The power is the rate of energy consumption; that is, P(t) = E′(t) Find the power over the interval 0 ≤ t ≤ 24.

Evaluate and simplify y'.

y = 2x^√2

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).