How do you find the derivative of #cos^2(x^2-2)#?

Answer 1

#d/dxcos^2(x^2-2) = 2x (sin(4-2x^2)) #

First of all define: #f(x) = cos(x^2-2)# First use the chain rule to find the derivative of f(x): #f'(x)= -sin(x^2-2) (2x)#
Next split #cos^2(x^2-2)# into #cos(x^2-2) cos(x^2-2)#. In other words: #f(x)^2 = f(x)f(x)#
Now the product rule can be used to find the derivative of f(x)^2: #f'(x)^2 = f'(x)f(x) + f(x)f'(x)# #f'(x)^2 = (-sin(x^2-2) (2x))(cos(x^2-2)) + (cos(x^2-2))(-sin(x^2-2) (2x)) #
#f'(x)^2 = 2 (cos(x^2-2))(-sin(x^2-2) (2x)) # #f'(x)^2 = -4x (cos(x^2-2)sin(x^2-2)) #
Using the trigonometric identity: #cos(x)sin(x) = 1/2 sin(2x)#
We finally gain: #f'(x)^2 = -4x (1/2(sin(2(x^2-2)) # #f'(x)^2 = -2x (sin(2x^2-4)) #
SIn(x) is an odd function, so it's true that: sin(x) = -sin(-x) Which means we can also write the answer as: #f'(x)^2 = 2x (sin(4-2x^2)) # Which removes the minus sign from the front, making the expression tidier.
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Answer 2

To find the derivative of ( \cos^2(x^2-2) ), you can use the chain rule and the derivative of the cosine function. The chain rule states that if ( f(x) = g(h(x)) ), then ( f'(x) = g'(h(x)) \cdot h'(x) ).

First, differentiate the outer function ( \cos^2(u) ) with respect to its inner function ( u ), which is ( x^2 - 2 ). Then, differentiate the inner function ( x^2 - 2 ) with respect to ( x ).

So, the derivative of ( \cos^2(x^2-2) ) with respect to ( x ) is:

[ \frac{d}{dx}[\cos^2(x^2-2)] = -2\cos(x^2-2)\sin(x^2-2) \cdot (2x) ]

Simplified, it becomes:

[ -4x\cos(x^2-2)\sin(x^2-2) ]

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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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