# What is the derivative of (x^2)(sinx)(tanx) without using the chain rule?

You can manipulate your function remembering that:

and

and get:

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Use the product rule for three factors:

We ca rewrite the middle term more simply, and we may choose to rewrite the third term:

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To find the derivative of (x^2 \cdot \sin(x) \cdot \tan(x)) without using the chain rule, you can use the product rule and trigonometric identities. First, expand the expression:

[y = x^2 \cdot \sin(x) \cdot \tan(x)]

Now, apply the product rule:

[y' = (x^2)' \cdot \sin(x) \cdot \tan(x) + x^2 \cdot (\sin(x))' \cdot \tan(x) + x^2 \cdot \sin(x) \cdot (\tan(x))']

Differentiate each term using basic differentiation rules and trigonometric identities:

[(x^2)' = 2x] [(\sin(x))' = \cos(x)] [(\tan(x))' = \sec^2(x)]

Substitute the derivatives back into the expression:

[y' = 2x \cdot \sin(x) \cdot \tan(x) + x^2 \cdot \cos(x) \cdot \tan(x) + x^2 \cdot \sin(x) \cdot \sec^2(x)]

This is the derivative of (x^2 \cdot \sin(x) \cdot \tan(x)) without 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.

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