http://www.ck12.org Chapter 2. Derivatives
f′(x) =x^2 (−sinx)+ 2 xcosx+cosx
=−x^2 sinx+ 2 xcosx+cosx.Example 3:
Finddy/dxify= 1 −costanxx. What is the slope of the tangent line atx=π/3?
Solution:
Using the quotient rule and the formulas above, we obtain
dy
dx=( 1 −tanx)(−sinx)−(cosx)(−sec^2 x)
( 1 −tanx)^2
=−sinx+tanxsinx+cosxsec(^2) x
( 1 −tanx)^2
=−sinx+( 1 tan−tanxsinx)x 2 +secx
To calculate the slope of the tangent line, we simply substitutex=π/3:
dy
dx
∣∣
∣∣
∣x=π 3 =−sin(π 3 )+tan(π 3 )sin(π 3 )+sec(π 3 )
( 1 −tan(π 3 ))^2.We finally get the slope to be approximately
dy
dx∣∣
∣∣
∣x=π 3 =^4.^9.Example 4:
Ify=secx, findy′′(π/ 3 ).
Solution:
y′=secxtanx
y′′=secx(sec^2 x)+(secxtanx)tanx
=sec^3 x+secxtan^2 x.Substituting forx=π/3,
y′′=sec^3(π
3)
+sec(π
3)
tan^2(π
3)
= ( 2 )^3 +( 2 )(
√
3 )^2
= 8 +( 2 )( 3 )
= 14.
Thusy′′(π/ 3 ) =14.