How do you differentiate #f(x)= (2x+1)(x^3+x^2) # using the product rule?
The product rule states that
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To differentiate ( f(x) = (2x+1)(x^3+x^2) ) using the product rule:
- Identify the two functions being multiplied together: ( u = 2x + 1 ) and ( v = x^3 + x^2 ).
- Apply the product rule, which states that ( (uv)' = u'v + uv' ).
- Find the derivatives of ( u ) and ( v ): ( u' = 2 ) (the derivative of ( 2x + 1 ) with respect to ( x )) ( v' = 3x^2 + 2x ) (the derivative of ( x^3 + x^2 ) with respect to ( x )).
- Substitute the derivatives and original functions into the product rule formula: ( f'(x) = (2)(x^3 + x^2) + (2x + 1)(3x^2 + 2x) ).
- Simplify the expression: ( f'(x) = 2x^3 + 2x^2 + 6x^3 + 4x^2 + 3x^2 + 2x ).
- Combine like terms: ( f'(x) = 8x^3 + 9x^2 + 2x ).
So, the derivative of ( f(x) = (2x+1)(x^3+x^2) ) with respect to ( x ) is ( f'(x) = 8x^3 + 9x^2 + 2x ).
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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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