All about the ancient tribes
A piecewise function is differentiable at a point if both of the pieces have derivatives at that point, and the derivatives are equal at that point. In this case, Sal took the derivatives of each piece: first he took the derivative of x^2 at x=3 and saw that the derivative there is 6.
Lesson 2.6: Differentiability: A function is differentiable at a point if it has a derivative there. Example 1: If f(x) is differentiable at x = a, then f(x) is also continuous at x = a. f(x) − f(a) (f(x) − f(a)) = lim. (x − a) · f(x) − f(a) x − a This is okay because x − a = 0 for limit at a. (x − a) lim. f(x) − f(a)
A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. Generally the most common forms of non- differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x.
No. A function with a removable discontinuity at the point is not differentiable at since it’s not continuous at. Continuity is a necessary condition.
Saying a function f is continuous when x=c is the same as saying that the function’s two-side limit at x=c exists and is equal to f(c).
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
A function is differentiable at a point when there’s a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.
When a function is differentiable it is also continuous. But a function can be continuous but not differentiable. For example the absolute value function is actually continuous ( though not differentiable ) at x=0.
A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.
denoted by R f ′ (c), are finite and equal. (ii) The function y = f (x) is said to be differentiable in the closed interval [a, b] if R f ′ (a) and L f ′ (b) exist and f ′ (x) exists for every point of (a, b).
Derivative Rules
Common Functions | Function | Derivative |
---|---|---|
Exponential | e^{x} | e^{x} |
a ^{x} | ln(a) a ^{x} | |
Logarithms | ln( x ) | 1/ x |
log_{a}( x ) | 1 / ( x ln(a)) |
what is the limit. The limit is what value the function approaches when x (independent variable) approaches a point. takes only positive values and approaches 0 (approaches from the right), we see that f(x) also approaches 0. exist at corner points.
Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain.