Academia Aeternum
तमसो मा ज्योतिर्गमय
Wisdom · Knowledge · Eternity
From ε–δ Intuition to Second Derivatives — Every Exercise Fully Solved
9 exercise files · 155 total questions
\(\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)\ \Rightarrow\ \text{continuous at }a\)\(\dfrac{d}{dx}(e^x)=e^x;\quad \dfrac{d}{dx}(\ln x)=\dfrac{1}{x}\)\(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\ \text{(parametric)}\)\(\dfrac{d}{dx}\big(f(g(x))\big) = f'(g(x))\cdot g'(x)\ \text{(chain rule)}\)Step 1 — Continuity check: compute LHL, RHL, and f(a) separately; all three must match. Step 2 — Implicit differentiation: differentiate both sides w.r.t. x, treat y as a function of x, then isolate dy/dx. Step 3 — Logarithmic differentiation: take ln of both sides first whenever the exponent itself contains x. Step 4 — Parametric: never eliminate the parameter unless asked — compute dy/dt and dx/dt separately and divide. Step 5 — Second derivative: differentiate dy/dx again, applying quotient/chain rule as needed.
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