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Purplepixie
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I would like to know how to differentiate |sin(t)| to obtain d(|sin(t)|)/d(t). Thank you!
Yes, the derivative of |x| can be written as \(\displaystyle \frac{x}{|x|}\) for x non-zero and does not exist at x= 0. Do you see why? It is not sufficient to memorize formulas. You need to understand why they are true!Purplepixie said:Thank you Country Boy, this is what I used: https://proofwiki.org/wiki/Derivative_of_Absolute_Value_Function
The derivative of |x| is not defined at x=0 because the function is not differentiable at this point. This means that the slope of the tangent line at x=0 does not exist, and therefore the derivative cannot be computed.
No, the derivative of |x| cannot be approximated at x=0 because the function is not continuous at this point. This means that the limit of the difference quotient, which is used to approximate the derivative, does not exist.
The geometric interpretation of the derivative of |x| at x=0 is that there is a sharp corner or "cusp" at this point. This is because the function changes direction abruptly at x=0, making it impossible to draw a unique tangent line.
Yes, the derivative of |x| is defined at all points except x=0. This is because the function is differentiable everywhere except at points where it is not continuous or has a sharp corner.
The derivative of |x| is related to the derivative of x^2 in that they have the same derivative at all points except x=0. This is because the function |x| can be rewritten as x^2 for all values of x except x=0. At x=0, the derivative of x^2 is 0, while the derivative of |x| does not exist.