3. Techniques of Differentiation

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

71. y = log₂(5θ)

10–19. Derivatives Find the derivatives of the following functions.

f(x) = ln(3 sin² 4x)

Calculate the derivative of the following functions.

y = sin (4x³ + 3x +1)

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

83. y = 3^(log₂ t)

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

a. On what day is the temperature increasing the fastest?

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9–61. Evaluate and simplify y'.

y = e^sin x+2x+1

Calculate the derivative of the following functions.

y = (e^x / x+1)⁸

Find the derivatives of the functions in Exercises 19–40.

q = sin(t / (√t + 1))

What is the derivative of y = e^kx?

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.

y = (2x + 1)⁵

Calculate the derivative of the following functions.

y = (1 - e^0.05x)^-1

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = cot(πu/10), u = g(x) = 5√x, x = 1

If r = sin(f(t)), f(0) = π/3, and f'(0) = 4, then what is dr/dt at t = 0?

27–76. Calculate the derivative of the following functions.

$\sqrt[3]{x^2+9}$

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.

d. ƒ(g(x)), x = 0