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How do you use the definition of a derivative to find the derivative of #f(x) = (x^2 + 2)^2#?

Answer 1

Charles Arocha

I found: #4x^3+8x#

Have a look:

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Answer 2

Christopher Allers

To find the derivative of the function f(x) = (x^2 + 2)^2 using the definition of a derivative, you apply the limit definition of the derivative:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Substitute the function f(x) into the definition:

f'(x) = lim(h->0) [(x + h)^2 + 2)^2 - (x^2 + 2)^2] / h

Expand the squares:

f'(x) = lim(h->0) [(x^2 + 2xh + h^2 + 2)^2 - (x^2 + 2)^2] / h

Expand the squared binomials:

f'(x) = lim(h->0) [(x^4 + 4x^3h + 4x^2h^2 + 4xh^3 + h^4 + 4x^2 + 8xh + 4 + 2 - x^4 - 4x^2 - 4)] / h

Simplify by canceling out terms:

f'(x) = lim(h->0) [4x^3h + 4x^2h^2 + 4xh^3 + h^4 + 8xh] / h

Now, as h approaches 0, all terms containing h will go to 0 except for the constant term:

f'(x) = 8x

So, the derivative of f(x) = (x^2 + 2)^2 with respect to x is f'(x) = 8x.

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Answer from HIX Tutor

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.