3. Techniques of Differentiation

Find the indicated derivative.

$f^{19}\left(x\right)$ of $f\left(x\right)=\cos x$

Find an equation of the line tangent to the following curves at the given value of x.

y = csc x; x = π/4

In Exercises 47 and 48, find an equation for

(a) the tangent line to the curve at P

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Find the derivative of the following functions.

y = cot x / (1 + csc x)

Derivatives

In Exercises 23–26, find dr/dθ.

r = (1 + sec θ) sin θ

9–61. Evaluate and simplify y'.

y=sin √cos² x+1

9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

Find the derivative of the function.

$f\left(x\right)=4x^2\sec x-\sqrt{x}$

9–61. Evaluate and simplify y'.

y = tan (sin θ)

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/4 cot x−1 / x−π/4

For what values of x does g(x) = x−sin x have a slope of 1?

9–61. Evaluate and simplify y'.

y = csc⁵ 3x

5-8. Use differentiation to verify each equation.

d/dx (tan³ x-3 tan x+3x) = 3 tan⁴x

Derivatives

In Exercises 1–18, find dy/dx.

y = (sec x + tan x)(sec x − tan x)

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

b. Find the particle’s velocity when t = 0, t = π/2, and t = π.