3. Techniques of Differentiation

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

y = (4x + 3)⁴(x + 1)⁻³

Find the derivative of the function.

$y=3{\cos }^4\theta$

[Technology Exercise]

Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula

y = 6(1 - t/12)² m.

a. Find the rate dy/dt (m/h) at which the tank is draining at time t.

15–48. Derivatives Find the derivative of the following functions.

y = In (x³+1)^π

Calculate the derivative of the following functions.

y = √x+√x+√x

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

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

g(t) = (1 + sin(3t) / (3 − 2t))⁻¹

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

y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

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.

b. f(x)g³(x), x = 0

Find the derivatives of the functions in Exercises 1–42.

s = cos⁴ (1 - 2t)

[Technology Exercise]

Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula

y = 6(1 - t/12)² m.

b. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy/dt at these times?

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Calculate the derivative of the following functions.

y = e^2x(2x-7)⁵

{Use of Tech} Cell population The population of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.

e. Graph the growth rate. When is it a maximum and what is the population at the time that the growth rate is a maximum? 

A general proof of the Chain Rule Let f and g be differentiable functions with h(x)=f(g(x)). For a given constant a, let u=g(a) and v=g(x), and define H (v) = <1x1 matrix>

c. Show that h′(a) = lim x→a ((H(g(x))+f′(g(a)))⋅g(x)−g(a)/x−a).