Differentiability Formula Sheet

Differentiability of a function at a point means that the function has a well-defined derivative at that point, indicating the existence of a unique tangent line to the function's graph at that point.

Neetesh Kumar | May 29, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Right & Left hand Derivatives
• 3. Important Points to Note
• 4. Derivability over an Interval

1. Differentiability:

The derivative of a function $f$ is a function; this function is denoted by symbols such as $f'(x)$, $\frac{df}{dx}$, $\frac{d[f(x)]}{dx}$ or $\frac{df(x)}{dx}$.

The derivative evaluated at a point $a$ can be written as: $f'_+(a)$, $\frac{df(a)}{dx}$, $f'(x)$, etc.

2. Right Hand & Left Hand Derivatives:

(a) Right-Hand Derivative:

The right hand derivative of $f(x)$ at $x = a$, denoted by $f'_+ (a)$, is defined as:

$f'_+ (a) = \lim\limits_{x\to0^+} \frac{{f(a + h) - f(a)}}{h}$, provided the limit exists and is finite.

(b) Left-Hand Derivative:

The left hand derivative of $f(x)$ at $x = a$, denoted by $f'_- (a)$, is defined as:

$f'_- (a) = \lim\limits_{x\to0^-} \frac{{f(a - h) - f(a)}}{-h}$, provided the limit exists and is finite.

(c) Derivability of Function at a Point:

If $f'_+ (a) = f'_- (a) =$ finite quantity, then $f(x)$ is said to be derivable or differentiable at $x = a$. In such case, $f'_+ (a) = f'_- (a) = f'(a)$, and it is called the derivative or differential coefficient of $f(x)$ at $x = a$.

Note:

• All polynomial, trigonometric, inverse trigonometric, logarithmic, and exponential functions are continuous and differentiable in their domains, except at endpoints.
• If f(x) & g(x) are derivable at x = a then the functions f(x) + g(x), f(x) – g(x), f(x). g(x) will also be derivable at x =a & if g(a) $\ne$ 0 then the function $\frac{f(x)}{g(x)}$ will also be derivable at x = a.

3. Important Note:

(a) Let $f'_+ (a) = p$ and $f'_- (a) = q$

Case(i): When $p$ and $q$ are finite:
If $p$ and $q$ are finite (whether equal or not), then $f$ is continuous at $x = a$ but the converse is NOT necessarily true.

• $p = q$ $\Rightarrow$ $f$ is differentiable at $x = a$ implies $f$ is continuous at $x = a$
  Example: $y = x^3 = f(x)$. $f^- (0) = 0$ and $f^+ (0) = 0$ implies $f'(0) = 0$,
  here, the x-axis is tangent to the curve at $x = 0$.
• $p \ne q$ $\Rightarrow$ $f$ is not differentiable at $x = a$, but $f$ is still continuous at $x = a$. In this case, we have a sudden change in the direction of the function's graph at x = a. This point is called a corner point of the function. At this point, there is no tangent to the curve.

Case(ii): When $p$ or $q$ may not be finite:
In this case, $f$ is not differentiable at $x = a$ and nothing can be concluded about the continuity of the function at $x = a$.

• Corner: If f is continuous at x = a with RHD and LHD at x = a, both are finite but not equal, or exactly one of them is infinite, then the point x = a is called a corner point, and at this point, the function is not differentiable but continuous.

• Cusp: If f is continuous at x = a and one of RHD, LHD at x = a, approaches to $\infty$ and the other approaches to –$\infty$, then the point x = a is called a cusp point. At the cusp point, we have a vertical tangent; at this point, the function is not differentiable but continuous. We can observe that the cusp is sharper than the corner point.

(b) Geometric Interpretation of Differentiability:

• If the function y = f(x) is differentiable at x = a, then a unique tangent can be drawn to the curve y = f(x) at P(a, f(a)) & f'(a) represent the slope of the tangent at point P.

• If LHD and RHD are finite but unequal, it geometrically implies a sharp corner at x = a. e.g. f(x) = |x| is continuous but f(x)=|x| not differentiable at x = 0. A sharp corner is seen at x = 0 in the f(x) = |x| graph.

(c) Vertical Tangent:

If y = f(x) is continuous at x = a and $\lim\limits_{x\to a}|f'(x)|$ approaches to $\infty$, then y = f(x) has a vertical tangent at x = a. If a function has a vertical tangent at x = a then it is non-differentiable at x = a.

4. Derivability Over an Interval:

(a) $f(x)$ is said to be derivable over an open interval $(a, b)$ if it is derivable at each and every point of the open interval $(a, b)$.

(b) $f(x)$ is said to be derivable over the closed interval $[a, b]$ if:

• $f(x)$ is derivable in $(a, b)$ and
• For the points $a$ and $b$, $f'_+ (a)$ and $f'_- (b)$ exist.

Additional Point to Note down:

• If $f(x)$ is differentiable at $x = a$ and $g(x)$ is not differentiable at $x = a$, then the product function $F(x) = f(x) \cdot g(x)$ can still be differentiable at $x = a$.
• If $f(x)$ and $g(x)$ are both not differentiable at $x = a$, then the product function $F(x) = f(x) \cdot g(x)$ can still be differentiable at $x = a$.
• If $f(x)$ and $g(x)$ are both non-derivable at $x = a$, then the sum function $F(x) = f(x) + g(x)$ may be a differentiable function.
• If $f(x)$ is derivable at $x = a$ then $f'(x)$ is continuous at $x = a$.
• The sum or difference of a differentiable and a non-differentiable function is always non-differentiable.

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