Home > Calculus > Relationship between First and Second Derivatives of a Function

What is the second derivative of #f(x)= sec^2x#?

Answer 1

Samantha Adkison

#f''(x)=4tan^2xsec^2x+2sec^4x#

To find the first derivative, we will have to use the chain rule on the second power.

Use the rule that #d/dx(u^2)=2u*u'#.

Thus, we see that

#f'(x)=2secx*d/dx(secx)#

#f'(x)=2secx*secxtanx#

#f'(x)=2sec^2xtanx#

To find the second derivative, we will have to use the product rule.

#f''(x)=2tanxd/dx(sec^2x)+2sec^2xd/dx(tanx)#

Note that we already know that #d/dx(sec^2x)=2sec^2xtanx# and that #d/dx(tanx)=sec^2x#.

This gives us

#f''(x)=2tanx(2sec^2xtanx)+2sec^2x(sec^2x)#

#f''(x)=4tan^2xsec^2x+2sec^4x#

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Answer 2

Christopher Allers

The second derivative of ( f(x) = \sec^2(x) ) is ( f''(x) = 2 \sec^2(x) \tan(x) ).

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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.