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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Section
BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

13–40 Calculate y ' .

y = tan x 1 + cos x

To determine

To find: The derivative of y

Explanation

Formula:

i.

Sum rule,  ddxfx+gx=ddx(fx+ddx(g(x)

ii.

Quotient rule,  ddxfxgx=gx*ddx(fx- fx*ddx(gxgx2

iii.

Trigonometric  rules,ddx(tanx)=sec2x  and ddx(cosx)= -sinx

iv.

Constant rule, ddxc= 0

v.

tanx=sinxcosx, cosx=1secx,sin2x+cos2x=1

vi.

a2-b2=a+b(a-b)

Given:

y=tanx1+cosx

Calculation:

Consider the given function

y=tanx1+cosx

Differentiate with respect to x on both sides

ddxy=ddxtanx1+cosx

Applying the quotient rule of differentiation

y'=(1+cosx)*ddx(tanx)-tanx*ddx(1+cosx)(1+cosx )2

By applying the trigonometric and sum rule of differentiation

y'=(1+cosx)*sec2x-tanx ddx1+ddxcosx(1+cosx)

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