How do you differentiate #g(x) =x^2tanx# using the product rule?

Answer 1

#g'(x)=2xtanx+x^2sec^2x#

The product rule states that the derivative of #uv#, where #u# and #v# are functions of #x#, is #u'v+uv'#. In other words, it's the derivative of one function times the other plus the derivative of the other function times the other.
We solve these problems by first finding the derivatives of each piece. In this case, we have #u=x^2# and #v=tanx# (note that #v# can be #x^2# and #u# can be #tanx#, it really doesn't matter). #d/dxx^2=2x# #d/dxtanx=sec^2x#
Now we substitute #u# and #v# and their derivatives into the product rule formula: #g'(x)=(x^2)'(tanx)+(x^2)(tanx)'# #g'(x)=2xtanx+x^2sec^2x#
We could rewrite this in other ways, like #xsecx(2sinx+secx)#, but they all mean the same thing. If your instructor prefers writing it using only sines and cosines, with the help of identities, you can express the result as #(xsin2x+x^2)/cos^2x#.
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Answer 2

g'(x) = # x^2sec^2x + 2xtanx #

applying the rule of the product:

g'(x) = f(x).h'(x) + h(x).f'(x) is the result if g(x) = f(x).h(x).

here let f(x) #=x^2color(black)(" and ") h(x)= tanx #
hence g'(x)# = x^2 d/dx(tanx) + tanx d/dx(x^2) #
# = x^2 (sec^2x) + tanx(2x) = x^2sec^2x + 2xtanx #
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Answer 3

To differentiate ( g(x) = x^2 \tan(x) ) using the product rule, follow these steps:

  1. Identify the two functions within the product: ( f(x) = x^2 ) and ( h(x) = \tan(x) ).
  2. Differentiate ( f(x) ) with respect to ( x ) to get ( f'(x) ).
  3. Differentiate ( h(x) ) with respect to ( x ) to get ( h'(x) ).
  4. Apply the product rule: ( g'(x) = f(x)h'(x) + f'(x)h(x) ).
  5. Substitute the derivatives and the original functions back into the formula to find ( g'(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.

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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