How do you find the derivative of #y=cos(x-1)# ?

Answer 1
One uses the chain rule, which states that for a function #y(x) = g(h(x)),# the derivative #dy/dx = h'(x)g'(h(x)) dx#
For this question, we must utilize the Chain rule. In this particular example, #g(h(x)) = cos(x-1), h(x)=x-1# The derivative #d/dx(x+c) = d/dx(x) + d/dx(c) = 1 + 0 = 1#, so #d/dx(h(x)) = d/dx(x-1) = 1#.
Regarding #g(h(x))#, since we are already accounting for the derivative of #h(x)#, simply find the derivative for #g(x)#. We know that the derivative of the cosine function is the negative sine function, from our knowledge of trigonometric function derivatives. Thus, #d/dx(cos x) = -sin(x)#.
NOTE: If we had not accounted for the derivative of h(x), we might instead use substitution, defining a variable #u# such that #u=h(x)#, and worked from there. However, we have accounted for #h'(x)# so this is currently unnecessary.

With our answers in hand, we apply the chain rule formula.

#dy/dx = h'(x)g'(h(x))dx = (1)(-sin(x-1))dx = -sin(x-1)dx = -sin(x-1)#
#dy/dx = -sin(x-1)#
Psykolord: I'm not comfortable seeing Leibniz and Lagrange on the same side of the equal sign. It's okay for integrals, but I'm not sure it's okay for derivatives. It's certainly doesn't look correct on your second last line: #-sin(x-1)dx=-sin(x-1)#. Please check your notes to verify that your solution is correct.
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Answer 2

To find the derivative of ( y = \cos(x-1) ), you can apply the chain rule. The derivative is given by:

[ \frac{dy}{dx} = -\sin(x-1) ]

This is because the derivative of cosine is negative sine, and according to the chain rule, you multiply by the derivative of the inner function, which is ( 1 ).

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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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