3. Techniques of Differentiation

Find the derivative of the following functions.

y = x sin x

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

d/dx (xf(x) / g(x)) |x=4

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.

b. Find an equation of the line tangent to y = h(x) at x=2.

Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).

f(x) = 1/x

Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

d/dx (xg(x)) | x=2

Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

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Find the derivatives with respect to x of the following combinations at the given value of x.

c. f(x) / (g(x) + 1), x = 1

Find and simplify the derivative of the following functions.

f(x) = e^x(x³ − 3x² + 6x − 6)

Find the derivatives of the functions in Exercises 17–28.

y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))

Population growth Consider the following population functions.

a. Find the instantaneous growth rate of the population, for t≥0.

p(t) = 600 (t²+3/t²+9)

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

a. Find L′(1) and interpret the meaning of this value.

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

c. ƒ(x) , x = 1

g(x) + 1