3. Techniques of Differentiation

In Exercises 41–58, find dy/dt.

y = 4 sin(√(1 + √t))

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = sin u, u = x − cos x

Finding Derivative Values

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

f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1

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

$y={(x^2+2x+7)}^8$

Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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

f. √f(x), x = 2

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

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

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𝓻 = √2θ sinθ

The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 - t)². How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

Calculate the derivative of the following functions.

y = (1 + 2 tan u)^4.5

Find f′(x) if f(x) = 15e^3x.

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

f(x) = √(7 + x sec x)

Evaluate and simplify y'.

y = (3t²−1 / 3t²+1)^−3

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

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

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𝔂 = ( √ x )²

( 1 + x )