- Chapter 5
- Calculus
- 24 questions
Continuity and Differentiability
Continuity, differentiability, the chain rule, implicit and parametric differentiation, logarithmic differentiation and second-order derivatives.
Continuity asks whether a function's graph can be drawn without lifting the pen. Differentiability asks something stricter: whether the graph is smooth enough to have a well-defined tangent at a point. Every differentiable function is continuous, but the reverse fails — |x| at the origin is the standard witness.
Most of the chapter is technique: the chain rule, and the ways of differentiating functions that are not written in the convenient form y = f(x). Implicit, parametric and logarithmic differentiation are each answers to that same difficulty.
What you should be able to do
- Test a function for continuity at a point and on an interval, including piecewise-defined functions.
- Explain the relationship between continuity and differentiability, and give a counter-example to the converse.
- Apply the chain rule to composite functions.
- Differentiate implicit, parametric, exponential, logarithmic and inverse trigonometric functions.
- Use logarithmic differentiation for products, quotients and variable exponents.
- Find second-order derivatives.
Topics
7 topics
- Continuity
Continuity at a point and on an interval, and the algebra of continuous functions.
- Continuity at a point
- Continuity of piecewise functions
- Algebra of continuous functions
- Differentiability and the Chain Rule
Differentiability, its relationship to continuity, and derivatives of composite functions.
- Derivatives of Inverse Trigonometric Functions
Differentiating, and, and composites built from them.
- Implicit Differentiation
Differentiating relations that are not solved for.
- Exponential and Logarithmic Functions
Derivatives of exponential and logarithmic functions, and logarithmic differentiation.
- Derivatives of exponential functions
- Derivatives of logarithmic functions
- Logarithmic differentiation
- Derivatives in Parametric Form
Finding when and are each given in terms of a parameter.
- Second Order Derivatives
Computing, including for implicit and parametric functions.
Formulas
10 items · All formulas
Continuity at a point
\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = f(a)All three quantities must exist and be equal.
Derivative from first principles
f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}Chain rule
\dfrac{d}{dx}\,f(g(x)) = f'(g(x)) \cdot g'(x)Differentiate the outer function, then multiply by the derivative of the inner.
Product rule
(uv)' = u'v + uv'Quotient rule
\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^{2}}- Valid when
- v \neq 0. Note the order in the numerator — it is not symmetric.
Derivatives of inverse trigonometric functions
\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1-x^{2}}}, \quad \dfrac{d}{dx}\cos^{-1}x = \dfrac{-1}{\sqrt{1-x^{2}}}, \quad \dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1+x^{2}}- Valid when
- |x| < 1 for the first two.
Exponential and logarithmic derivatives
\dfrac{d}{dx}e^{x} = e^{x}, \quad \dfrac{d}{dx}a^{x} = a^{x}\ln a, \quad \dfrac{d}{dx}\ln x = \dfrac{1}{x}- Valid when
- a > 0, a \neq 1; x > 0 for the logarithm.
Logarithmic differentiation
y = f(x)^{g(x)} \;\Rightarrow\; \dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{d}{dx}\left[g(x)\ln f(x)\right]For a variable base with a variable exponent.
- Valid when
- f(x) > 0.
Parametric differentiation
\dfrac{dy}{dx} = \dfrac{\;\dfrac{dy}{dt}\;}{\dfrac{dx}{dt}}- Valid when
- \dfrac{dx}{dt} \neq 0.
Second derivative of a parametric function
\dfrac{d^{2}y}{dx^{2}} = \dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right) \cdot \dfrac{1}{\;\dfrac{dx}{dt}\;}The extra factor \dfrac{dt}{dx} is essential — differentiating \frac{dy}{dx} with respect to t alone is not the second derivative.
Key points
7 items · All key points
Differentiability implies continuity, but continuity does not imply differentiability.
f(x) = |x| is continuous at 0 but has no derivative there.
For a piecewise function, always compute LHL, RHL and f(a) separately and compare all three.
A function that is discontinuous at a point cannot be differentiable there — say so and stop.
In logarithmic differentiation, the left side becomes \dfrac{1}{y}\dfrac{dy}{dx}, not \dfrac{dy}{dx}.
Always substitute y back in terms of x at the end of logarithmic or implicit differentiation.
For a sum such as x^{x} + x^{\sin x}, differentiate each term separately — \ln(a+b) does not simplify.
For parametric second derivatives, remember the final factor \dfrac{1}{dx/dt}.
\dfrac{d^2y}{dx^2} \neq \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right) — the chain rule demands one more step.
Common mistakes
6 items · All common mistakes
- Mistake
Checking only one side of a piecewise function when testing continuity.
InsteadCompute the left-hand limit, the right-hand limit and the function value, then compare all three.
WhyOne side often matches the function value, which feels like a complete check.
- Mistake
Concluding that a continuous function must be differentiable.
InsteadThe implication runs the other way only. Corners and cusps are continuous but not differentiable.
- Mistake
Writing \dfrac{d}{dx}(\ln y) = \dfrac{dy}{dx} instead of \dfrac{1}{y}\dfrac{dy}{dx}.
Insteady is a function of x, so the chain rule applies: \dfrac{d}{dx}(\ln y) = \dfrac{1}{y}\cdot\dfrac{dy}{dx}.
WhyThe chain rule is easy to forget when the inner function is written as a single letter.
- Mistake
Differentiating x^{x} as x \cdot x^{x-1} (power rule) or x^{x}\ln x (exponential rule).
InsteadNeither rule applies — the base and the exponent vary. Use logarithms: \dfrac{dy}{dx} = x^{x}(1 + \ln x).
WhyThe expression resembles both standard forms, so one of the two rules gets applied by reflex.
- Mistake
Omitting the factor \dfrac{1}{dx/dt} when finding a parametric second derivative.
Instead\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)\cdot\dfrac{dt}{dx}.
- Mistake
Leaving the answer in terms of y after implicit or logarithmic differentiation.
InsteadSubstitute the original expression for y to give the answer in x.