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Find the second derivative of #f(x)=x^4 tan x#?

Answer 1

Charles Arocha

#f''(x)=2x^4\sec^2x\tan x+8x^3\sec^2x+12x^2\tan x#

Given that

#f(x)=x^4\tan x#

Differentiating above function w.r.t. #x# using product rule as follows

#d/dxf(x)=d/dx(x^4\tan x)#

#f'(x)=x^4d/dx\tan x+\tan xd/dxx^4#

#=x^4\sec^2x+\tan x(4x^3)#

#=x^4\sec^2x+4x^3\tan x#

Differentiating above equation w.r.t. #x# using product rule as follows

#d/dxf'(x)=d/dx(x^4\sec^2x+4x^3\tan x)#

#f''(x)=x^4d/dx(sec^2x)+\sec^2xd/dx(x^4)+4x^3d/dx(\tan x)+4\tan xd/dx(x^3)#

#=x^4(2\sec x\sec x\tan x)+\sec^2x(4x^3)+4x^3(\sec^2x)+4\tan x(3x^2)#

#=2x^4\sec^2x\tan x+4x^3\sec^2x+4x^3\sec^2x+12x^2\tan x#

#=2x^4\sec^2x\tan x+8x^3\sec^2x+12x^2\tan x#

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Answer 2

Emma Aldridge

To find the second derivative of ( f(x) = x^4 \tan(x) ), follow these steps:

Find the first derivative ( f'(x) ) using the product rule. [ f'(x) = 4x^3 \tan(x) + x^4 \sec^2(x) ]
Now, differentiate ( f'(x) ) using the product rule again. [ f''(x) = 12x^2 \tan(x) + 4x^3 \sec^2(x) + 4x^3 \tan(x) \sec^2(x) + 2x^3 \tan(x) \sec^2(x) ]
Simplify the expression if necessary.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.