How To Check If A Function Is Differentiable

Have you ever wondered what makes a function distinguishable? A formal function is said to be distinguishable if its derivative exists at each point in its domain, but what does this mean? Its derivative is defined So as long as you can evaluate the derivative at every point on the curve, the function is distinguishable.

How to determine the ability to differentiate

By using limits and continuity! The definition of differentiability is shown below:

• f is differentiable over an open interval (a, b) if (lim _ {h rightarrow 0} frac {f (c + h) -f (c)} {h}) exists for every c in (a, b) .
• If f is differentiable, that is, if it exists (f^{prime}(c)) then f is continuous at c.

Since then, difference is when the slope of the tangent line is equal to the limit of the function at a given point. This directly suggests that for a function to be distinguishable it must be continuous and its derivative must also be continuous. )} {h}) does not exist, then we can conclude that f(x) is indistinguishable at x = 3 because it (f^{prime}(3)) does not exist. very important meaning – therefore all distinguishable functions must be continuous, but not all continuous functions are distinguishable! Therefore, the only way for the derivative to exist is if the function also exists (i.e. is continuous) on its domain. Thus, a discriminant function is also a continuous function, but just because a function is continuous does not mean that its derivative (i.e. slope of the tangent) is defined everywhere in the domain. graph(f(x) = |x|). We can easily see that the absolute value graph is continuous because we can draw the graph without picking up the pencil. left as well as right. This shows that the instantaneous rate of change at the peak is different (i.e. x = 0). So what should we do? We use one-sided limit and our definition of the derivative to determine if the slopes on the left and right sides are equal. start {equation} begin {array} {l} lim _ {h rightarrow 0 ^ {-}} frac {f (x + h ) -f (x)} {h} = lim _ {h rightarrow 0 ^ { -}} frac {(- (x + h)) – (- x)} {h} = lim _ {h rightarrow 0 ^ { -}} frac {-xh + x} {h} lim _ {h rightarrow 0 ^ {-}} frac {-h} {hbar} = lim _ {h rightarrow 0 ^ {-}} (- 1) = -1 lim _ {h rightarrow 0 ^ {+}} frac {f (x + h) -f (x)} {h} = lim _ {h rightarrow 0 ^ {+}} frac {((x + h)) – (x)} {h} = lim _ {h rightarrow 0 ^ {+ }} frac {x + hx} {h} lim _ {h rightarrow 0 ^ {+}} frac {h} {h} = lim _ {h rightarrow 0 ^ {+}} (1) = 1 end {array} end {method} And when we compare, we see that the slope of the left side is -1 and the slope of the right side is +1 so they disagree.Read more: How to tie a golden goosestring Hence, the function f( x) = | x | indistinguishable at x = 0. While the function is continuous, it is indistinguishable because the derivative is discontinuous everywhere, as seen in the graph below.

The discriminant ability of a function

So how do you know if a function is discriminable or not? The easiest way to determine the discriminability is to look at the graph of the function and check that it doesn’t contain any “problems” that cause the instantaneous rate of change to become unknown, that is:

• Cusp or Corner (turning around)
• Discontinuity (jump, point or infinite)
• Vertical tangent (unknown slope)

So armed with this knowledge, use the graph below to determine which numbers f(x) is indistinguishable from and why.

Example) What value of X is F(X) indistinguishable?

• At x = -8, 0 and 3 (not continuous)
• At x = -4 and 2 (vertex/angle)
• At x = -6.5 (vertical tangent)

See, that’s not too hard to figure out, is it?

Summary

So, in this video lesson, you will learn how to determine if a function is distinguishable graphically or by using left and right derivatives. In addition, you will also learn how to find the values ​​that make a function distinguishable.

Video Tutorial w/ Full Lesson & Detailed Example (Video)

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