“Differentiable is stronger”
Continuity and differentiability
Every differentiable function is continuous, but not the reverse. A sharp corner is continuous and has no single tangent.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Continuity and Differentiability formulas
This chapter supplies the rules used everywhere else in calculus. Learn the standard derivatives as a block, then the four rules, and the rest of the year's differentiation is routine.
16 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Continuity
| What | Formula | Symbols and notes |
|---|---|---|
| Continuous at a point | left limit = right limit = f(a) | all three must exist and agree |
| Differentiable implies continuous | but not the reverse | |x| is continuous at 0 and not differentiable there |
| Standard check | find the limits from both sides, then compare with the value | the method for every piecewise function |
Standard derivatives
| What | Formula | Symbols and notes |
|---|---|---|
| Power | d/dx (xn) = n xn−1 | the one used most |
| Trigonometric | d/dx (sin x) = cos x, d/dx (cos x) = −sin x | note the minus |
| More trigonometric | d/dx (tan x) = sec²x, d/dx (sec x) = sec x tan x | and cot gives −cosec²x |
| Exponential and log | d/dx (ex) = ex, d/dx (ln x) = 1/x | ex is its own derivative |
| General exponential | d/dx (ax) = ax ln a | reduces to ex when a = e |
| Inverse trigonometric | d/dx (sin−1x) = 1/√(1 − x²) | and d/dx (tan−1x) = 1/(1 + x²) |
Rules
| What | Formula | Symbols and notes |
|---|---|---|
| Product rule | (uv)' = u'v + uv' | differentiate one at a time |
| Quotient rule | (u/v)' = (u'v − uv')/v² | the numerator order matters |
| Chain rule | dy/dx = (dy/du)(du/dx) | for a function of a function |
| Implicit differentiation | differentiate both sides, then collect dy/dx | used when y is not isolated |
| Parametric form | dy/dx = (dy/dt)/(dx/dt) | when x and y are both given in terms of t |
| Logarithmic differentiation | take ln of both sides first | for forms like xx |
| Second derivative | d²y/dx² = d/dx(dy/dx) | not the square of the first derivative |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“u dash v plus u v dash”
The product rule
Differentiate the first times the second, plus the first times the derivative of the second. Say it aloud once and the order sticks.
“Low d-high minus high d-low, over low squared”
The quotient rule
The numerator order matters, and reversing it flips the sign of the whole answer.
“Take log for a power of x”
Differentiating xx
Neither the power rule nor the exponential rule applies when the base and index both vary. Take logarithms first.
Where marks go
The mistakes that cost the most.
- Using the power rule on ax. The variable is the index, so the exponential rule applies.
- Writing the second derivative as the square of the first.
- Forgetting the inner derivative in the chain rule, which is the most common single error in calculus.
- Reversing the numerator of the quotient rule.
- Assuming a continuous function must be differentiable everywhere.
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