280
Chapter 3 | Derivatives
Find the derivative of g ( x ) = cos x 4 x 2 . Solution By applying the quotient rule, we have
x )4 x 2 −8 x (cos x ) ⎛ ⎝ 4 x 2 ⎞ ⎠ 2
g ′( x ) = (−sin
.
Simplifying, we obtain
2 sin x −8 x cos x 16 x 4
g ′( x ) = −4 x
= − x sin x −2cos x 4 x 3 .
Find the derivative of f ( x ) = x cos x .
3.26
Example 3.41 An Application to Velocity
A particle moves along a coordinate axis in such a way that its position at time t is given by s ( t ) =2sin t − t for 0≤ t ≤2 π . At what times is the particle at rest?
Solution To determine when the particle is at rest, set s ′( t ) = v ( t ) =0. Begin by finding s ′( t ). We obtain s ′( t ) =2cos t −1, so we must solve 2cos t −1=0for 0≤ t ≤2 π . The solutions to this equation are t = π 3 and t = 5 π 3 . Thus the particle is at rest at times t = π 3
and t = 5 π
3 .
3.27
A particle moves along a coordinate axis. Its position at time t is given by s ( t ) = 3 t +2cos t for 0≤ t ≤2 π . At what times is the particle at rest?
Derivatives of Other Trigonometric Functions Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.
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