How do you differentiate #g(y) =(x^3 + x)(4x^2+5) # using the product rule?
Product rule states that for
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To differentiate ( g(y) = (x^3 + x)(4x^2+5) ) using the product rule, follow these steps:
- Identify the two functions being multiplied together: ( f(x) = x^3 + x ) and ( h(x) = 4x^2 + 5 ).
- Apply the product rule: ( g'(x) = f'(x)h(x) + f(x)h'(x) ).
- Find the derivatives of ( f(x) ) and ( h(x) ): ( f'(x) = 3x^2 + 1 ) and ( h'(x) = 8x ).
- Substitute these derivatives into the product rule formula.
- Simplify the expression.
The derivative of ( g(y) ) using the product rule is:
[ g'(x) = (3x^2 + 1)(4x^2 + 5) + (x^3 + x)(8x) ]
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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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