3. Techniques of Differentiation

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

𝓻 = ( sin θ )²

( cos θ - 1 )

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

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.

g. ƒ(x + g(x)), x = 0

Finding Derivative Values

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

f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

Finding Derivative Values

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

f(u) = u + 1/cos²u, u = g(x) = πx, x = 1/4

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

In Exercises 41–58, find dy/dt.

y = √(3t + (√2 + √(1 − t)))

In Exercises 41–58, find dy/dt.

y = tan²(sin³(t))

In Exercises 41–58, find dy/dt.

y = 4 sin(√(1 + √t))

In Exercises 41–58, find dy/dt.

y = (1/6)(1 + cos²(7t))³

In Exercises 41–58, find dy/dt.

y = (1 + tan⁴(t/12))³

In Exercises 41–58, find dy/dt.

y = ((3t − 4) / (5t + 2))⁻⁵

In Exercises 41–58, find dy/dt.

y = (t⁻³/⁴ sin(t))⁴/³

In Exercises 41–58, find dy/dt.

y = (t tan(t))¹⁰

In Exercises 41–58, find dy/dt.

y = (1 + cos(2t))⁻⁴