How do you differentiate #f(x)= tanx# twice using the quotient rule?

Answer 1

#dy/dx=sec^2x#

#(d^2y)/dx^2=2sec^2xtanx#

In order to start properly, let's just remember that #tanx=(sinx)/(cosx)# and now differentiate this quotient using the proper rule.
The quotient rule states that for a function #y=f(x)/g(x)#, #(dy)/(dx)=(f'(x)g(x)-f(x)g'(x))/g(x)^2#

Thus, in regards to your role:

#f(x)=sinx# #g(x)=cosx# #f'(x)=cosx# #g'(x)=-sinx#
#(dy)/(dx)=(cosxcosx-sinx(-sinx))/(cosx)^2=(cos^2x+sin^2x)/cos^2x#
From trigonometric identities, we know that #sin^2x+cos^2x=1#
#(dy)/(dx)=1/(cos^2x)=sec^2x#
To find the second derivative, use the chain rule, which states that for a function #y=f(g(x))#, #dy/dx=f'(g(x))g'(x)#.
First, rewrite #dy/dx=cos^-2x#

Next, in line with the chain rule:

#(d^2y)/dx^2=-2cos^-3xd/dx(cosx)#
#=>(-2(-sinx))/cos^3x#

Your understanding of the concluding response may differ.

#(d^2y)/dx^2=(2sinx)/cos^3x# #"or"# #(d^2y)/dx^2=2sec^2xtanx#
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Answer 2

To differentiate ( f(x) = \tan(x) ) twice using the quotient rule, first, differentiate ( f(x) ) once to find ( f'(x) ), then differentiate ( f'(x) ) to find ( f''(x) ).

The function ( f(x) = \tan(x) ) can be expressed as the quotient of two functions: ( f(x) = \frac{\sin(x)}{\cos(x)} ).

  1. First differentiation: Apply the quotient rule: [ f'(x) = \frac{\cos(x) \cdot \cos(x) - (-\sin(x)) \cdot \sin(x)}{\cos^2(x)} ] [ f'(x) = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} ] Since ( \cos^2(x) + \sin^2(x) = 1 ): [ f'(x) = \frac{1}{\cos^2(x)} ]

  2. Second differentiation: Apply the quotient rule to ( f'(x) = \frac{1}{\cos^2(x)} ): [ f''(x) = \frac{-2\sin(x)(-\sin(x))}{\cos^3(x)} ] [ f''(x) = \frac{2\sin^2(x)}{\cos^3(x)} ] Since ( \sin^2(x) = 1 - \cos^2(x) ): [ f''(x) = \frac{2(1 - \cos^2(x))}{\cos^3(x)} ] [ f''(x) = \frac{2 - 2\cos^2(x)}{\cos^3(x)} ]

    You can simplify this expression further, but this is the second derivative of ( f(x) = \tan(x) ) obtained using the quotient rule twice.

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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.

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.

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.

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.

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