What is the derivative of #(4x^8-sqrt(x))/(8x^4)#?
To begin, rewrite your function as follows:
This could be rephrased as
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To find the derivative of the function ((4x^8 - \sqrt{x})/(8x^4)), we can use the quotient rule. The quotient rule states that if (u(x)) and (v(x)) are differentiable functions, then the derivative of their quotient is ((u'v - uv')/(v^2)). Applying the quotient rule to the given function, we have:
[ f(x) = \frac{4x^8 - \sqrt{x}}{8x^4} ]
where (u(x) = 4x^8 - \sqrt{x}) and (v(x) = 8x^4). Now, differentiate (u(x)) and (v(x)) with respect to (x):
[ u'(x) = 32x^7 - \frac{1}{2\sqrt{x}} ]
[ v'(x) = 32x^3 ]
Now, apply the quotient rule:
[ f'(x) = \frac{(32x^7 - \frac{1}{2\sqrt{x}})(8x^4) - (4x^8 - \sqrt{x})(32x^3)}{(8x^4)^2} ]
Simplify this expression to get the derivative of the given function.
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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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