3. Techniques of Differentiation

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = e^4x²+1

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

_____

𝔂 = / x² + x

√ x²

Calculate the derivative of the following functions.

y = (2x⁶ - 3x³ + 3)²⁵

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = sin⁵x

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

In Exercises 41–58, find dy/dt.

y = (t tan(t))¹⁰

9–61. Evaluate and simplify y'.

y = (v / v+1)^4/3

Find dy/dt when x = 1 if y = x² + 7x − 5 and dx/dt = ¹/₃.

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{h\to 0}{\lim }\frac{\frac{1}{3{({(1+h)}^5+7)}^{10}}-\frac{1}{3{\left(8\right)}^{10}}}{h}$

In Exercises 41–58, find dy/dt.

y = √(3t + (√2 + √(1 − 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.

b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?

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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.

g. f(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

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

67. y = 7^(sec θ) ln 7"

In Exercises 41–58, find dy/dt.

y = sin²(πt − 2)