Using Symmetric Difference Quotient To Define Derivative – An Alternative Approach

Alternative form of the definition of the derivative

lim x->c. f(x)-f(c) / x-c

An alternative form of the definition of the derivative is:

$$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$$

This definition is derived by using the symmetric difference quotient, which takes the average of the slopes of secant lines on either side of the point in question. As the limit of $h$ approaches zero, this definition calculates the slope of the tangent line at the point $x$.

This formula is useful in situations where the function is not easily modified or where implementing the original definition of the derivative is challenging. The symmetric difference quotient approximates the derivative for small values of $h$ with a high degree of accuracy.

Both definitions are equivalent and can be used to calculate the derivative of a function, but the choice between them depends on the specific situation and the tools available.

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