Determine where the function f ( x ) f\left(x\right) f ( x ) is not differentiable. f ( x ) = 3 x + 2 f\left(x\right)=\frac{3}{x+2} f ( x ) = x + 2 3

in this problem, we are given the function f of x is equal to 3 over x plus 2, and we're asked to determine where the function f of x is not differentiable. Now, whenever we're trying to figure out where something is not differentiable, a good place to start is to first see whether or not the function is continuous for all x values. Now I see that we have this function right here, which is a rational function. And one of the things that I know is that there is a domain restriction for rational functions. The denominator cannot be equal to 0, the bottom of the fraction. Because whenever this bottom of the fraction is 0, that means we'd be dividing by 0, and that's something we cannot do in math. So we can say that x plus 2 cannot equal 0, and what we can do is solve this inequality for x. So I'm gonna go ahead and subtract 2 on both sides here, and that will get the 2's to cancel right over there, leaving me with x cannot equal negative 2. So this right here is a place where the function would break if we try to plug in negative 2 for x. And if we do that, we would actually end up having an asymptote in our graph. And recall that a function is not continuous whenever you have a junk, a hole, or an asymptote, which is a place where the function cannot exist. So in this case, we would say that at x equals negative 2, our function here is not continuous. And because it's not continuous, by default, it's not going to be differentiable at x equals negative 2. So this right here would be the solution, and that is where the function is not differentiable. So I hope you found this video helpful. Let's go ahead and get some more practice.

2. Intro to Derivatives

Differentiability

Business Calculus

2. Intro to Derivatives

Differentiability

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Multiple Choice

Determine where the function $f\left(x\right)$ is not differentiable.
$f\left(x\right)=\frac{3}{x+2}$

-2

Verified step by step guidance

Step 1: Recall that a function is not differentiable at points where it is either discontinuous or where the derivative does not exist (e.g., sharp corners, vertical tangents, or undefined points).

Step 2: Analyze the given function f(x) = 3 / (x + 2). Notice that the denominator (x + 2) cannot be zero, as division by zero is undefined. This means the function is undefined at x = -2.

Step 3: Since the function is undefined at x = -2, it is also discontinuous at this point. A function cannot be differentiable at a point where it is discontinuous.

Step 4: Verify that there are no other points where the function might fail to be differentiable. For rational functions like this one, differentiability issues typically arise only at points where the denominator is zero or where the function has a sharp corner or vertical tangent. In this case, the only issue is at x = -2.

Step 5: Conclude that the function f(x) is not differentiable at x = -2 due to the discontinuity caused by the denominator being zero at this point.

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Struggling with Business Calculus?

Determine where the function f(x)f\left(x\right)f(x) is not differentiable.f(x)=3x+2f\left(x\right)=\frac{3}{x+2}f(x)=x+23

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Differentiability practice set

Determine where the function $f\left(x\right)$ is not differentiable.
$f\left(x\right)=\frac{3}{x+2}$