3. Techniques of Differentiation

Find the derivative of the function.

$y=\frac{\sin \theta }{2+\cos \theta }$

Find the indicated derivative.

$d^{52}y/d{x}^{52}$ of $y=\cos x$

Find the indicated derivative.

$d^{1002}/d{x}^{1002}\left(\sin x\right)$

Consider the function

f(x) = { x² cos(2/x), x ≠ 0

0, x = 0

b. Determine f' for x ≠ 0.

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (csc x) = -csc x cot x

Derivatives

In Exercises 1–18, find dy/dx.

f(x) = x³ sin x cos x

Derivatives

In Exercises 27–32, find dp/dq.

p = (sin q + cos q) / cos q

Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ

Find the derivative of the function: $f\left(x\right)=cos\left(x^2\right)$. Which of the following is correct?

Differentiate the function: $f\left(t\right)={ln\left(t\right)}^{2}cos\left(t\right)$. Which of the following is the correct derivative $f^{\prime }\left(t\right)$?

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/2 cos x/x−π/2

Theory and Examples

The equations in Exercises 49 and 50 give the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.

s = 2 − 2 sin t

Find the derivative of the function.

$r=\csc x-3\sin x+\tan x$

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

Find the velocity of the oscillator, v(t) =y′(t).