How I do I prove the Chain Rule for derivatives?
Let us first note the following before moving on to the proof.
Evidence for the Chain Rule
based on the aforementioned observation,
According to the derivative's definition,
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To prove the Chain Rule for derivatives, you can use the definition of the derivative along with the limit definition. Let ( f(x) = g(h(x)) ), where ( g ) is differentiable at ( h(x) ) and ( h ) is differentiable at ( x ).

Start with the definition of the derivative of ( f ): [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]

Substitute ( f(x) = g(h(x)) ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{g(h(x) + h)  g(h(x))}{h} ]

Rewrite the numerator using the difference quotient: [ f'(x) = \lim_{h \to 0} \frac{g(h(x) + h)  g(h(x))}{h} \times \frac{h}{h} ]

Use the fact that ( h(x) ) is differentiable at ( x ): [ f'(x) = \lim_{h \to 0} \frac{g(h(x) + h)  g(h(x))}{h} \times \lim_{h \to 0} \frac{h}{h} ]

Simplify the limit of the second factor: [ f'(x) = \lim_{h \to 0} \frac{g(h(x) + h)  g(h(x))}{h} ]

Recognize that the first factor is the definition of the derivative of ( g ) at ( h(x) ): [ f'(x) = g'(h(x)) \times h'(x) ]
This proves the Chain Rule for derivatives.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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