3. Techniques of Differentiation

Calculate the derivative of the following functions.

y = tan e^x

Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?

Find the derivative of the function.

$f(x)=\(\sin\)^5(2x^3+1)$

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.

h. √(f²(x) + g²(x)), x = 2

In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.

y = (4 − 3x)⁹

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(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.

g. ƒ(x + g(x)), x = 0

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (f(f(x))) |x=4

Calculate the derivative of the following functions.

y = csc (t² + t)

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

65. y = (cos θ)^(√2)

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

𝔂 = (θ² + sec θ + 1)³

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 = 3x + 1

Calculate the derivative of the following functions.

y = cos^7/4(4x³)

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

69. y = 2^(sin 3t)