How do you differentiate #f(x)=sqrt(csc(1/x^3 ) # using the chain rule?
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To differentiate ( f(x) = \sqrt{\csc\left(\frac{1}{x^3}\right)} ) using the chain rule, follow these steps:
- Let ( u(x) = \frac{1}{x^3} ).
- Find ( u'(x) ), the derivative of ( u(x) ) with respect to ( x ).
- Let ( v(u) = \csc(u) ).
- Find ( v'(u) ), the derivative of ( v(u) ) with respect to ( u ).
- Use the chain rule: ( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} ).
- Substitute ( u ) back in terms of ( x ) and simplify the result.
The derivative ( \frac{df}{dx} ) will give you the result of differentiating ( f(x) ) with respect to ( x ) using the chain rule.
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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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