How to solve for the derivative given this information: #f(x+h)f(x) = 2 h x^2 + 3 h x  8 h^2 x + 8 h^2  4 h^3# ?
Divide both sides of the equation by h:
This is the definition of the derivative.
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To solve for the derivative using the provided information (f(x+h)f(x) = 2hx^2 + 3hx  8hx^2 + 8h^2  4h^3):

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

Substitute the given expression for (f(x+h)f(x)) into the difference quotient: [f'(x) = \lim_{h \to 0} \frac{2hx^2 + 3hx  8hx^2 + 8h^2  4h^3}{h}]

Simplify the expression by canceling out common terms and factoring out (h): [f'(x) = \lim_{h \to 0} \frac{(2x^2  8x^2)h + (3x)h + (8h^2  4h^3)}{h}] [f'(x) = \lim_{h \to 0} \frac{(6x^2 + 3x)h + (8h^2  4h^3)}{h}] [f'(x) = \lim_{h \to 0} (6x^2 + 3x) + \frac{(8h^2  4h^3)}{h}]

Evaluate the limit as (h) approaches 0: [f'(x) = 6x^2 + 3x + \lim_{h \to 0} (8h  4h^2)]

The limit term (\lim_{h \to 0} (8h  4h^2)) becomes 0 when (h) tends to 0.

Therefore, the derivative (f'(x)) simplifies to: [f'(x) = 6x^2 + 3x]
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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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