3. Techniques of Differentiation

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

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

𝔂 = 3 .

(5x² + sin 2x)³/²

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

𝔂 = (θ² + sec θ + 1)³

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

𝔂 = 2 tan² x - sec² x

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

s = cos⁴ (1 - 2t)

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

s = (sec t + tan t)⁵

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

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𝓻 = √2θ sinθ

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

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𝓻 = sin √ 2θ

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

𝔂 = x² sin² (2x²)

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

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𝔂 = ( √ x )²

( 1 + x )

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

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𝔂 = / x² + x

√ x²

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