Chapter 3 | Derivatives
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differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative. • We can use a formula to find the derivative of y = ln x , and the relationship log b x = ln x ln b our differentiation formulas to include logarithms with arbitrary bases. • Logarithmic differentiation allows us to differentiate functions of the form y = g ( x ) f ( x ) or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating. allows us to extend
CHAPTER 3 REVIEW EXERCISES True or False ? Justify the answer with a proof or a counterexample. 367. Every function has a derivative.
382. Third derivative of y = (3 x +2) 2
383. Second derivative of y =4 x + x 2 sin( x )
368. A continuous function has a continuous derivative.
Find the equation of the tangent line to the following equations at the specified point. 384. y =cos −1 ( x )+ x at x =0
369. A continuous function has a derivative.
370. If a function is differentiable, it is continuous.
385. y = x + e x − 1
x at x =1
Use the limit definition of the derivative to exactly evaluate the derivative. 371. f ( x ) = x +4
Draw the derivative for the following graphs. 386.
372. f ( x ) = 3 x
Find the derivatives of the following functions. 373. f ( x ) =3 x 3 − 4 x 2
3
⎛ ⎝ 4− x 2
⎞ ⎠
374. f ( x ) =
387.
375. f ( x ) = e sin x
376. f ( x ) = ln( x +2)
377. f ( x ) = x 2 cos x + x tan( x )
378. f ( x ) = 3 x 2 +2
379. f ( x ) = x
−1 ( x )
The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by w ( t ) = 1.9 + 2.9cos ⎛ ⎝ π 6 t ⎞ ⎠ , where t is measured in hours after midnight, and the height is measured in feet. 388. Find and graph the derivative. What is the physical meaning?
4 sin
380. x 2 y = ⎛
⎝ y +2 ⎞
⎠ + xy sin( x )
Find the following derivatives of various orders. 381. First derivative of y = x ln( x )cos x
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