3. Techniques of Differentiation

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

f(x) = 5x³

Derivatives Find and simplify the derivative of the following functions.

g(t) = t³+3t²+t / t³

21–30. Derivatives

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

f(t) = 1/√t; a=9, 1/4

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at

b. x=2

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)

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

y = 12s³-8s²+12s/4s

Let f(x) = x² - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 0.

Let f(x) = x² - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 2.

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(w) = w³-w/w

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.

Derivatives of even and odd functions Recall that f is even if f(−x) = f(x), for all x in the domain of f, and f is odd if f(−x) = −f(x) for all x in the domain of f.

b. If f is a differentiable, odd function on its domain, determine whether f' is even, odd, or neither.

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

Find and simplify the derivative of the following functions.

f(x) = 3x^-9