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Understanding Calculus for Finance: Derivatives, Integrals, and Continuous Compounding

A ground-up, finance-first introduction to the calculus that quant work actually uses. Starts with the idea of a rate of change using Nifty 50 price paths, builds derivatives as sensitivities (duration, delta, marginal cost), then integrals as accumulation (total return, expected payoff, area under a distribution), and finishes with the exponential function and continuous compounding that underpin discounting, log returns, and Black-Scholes. Every concept is anchored in NSE and BSE data, INR examples, and screener.in style numbers rather than abstract textbook problems.

MODULES

6

DURATION

4 Hours

TRACK

Quantitative Finance

Access Level

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Actionable Exercises

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Curriculum Breakdown

Chapter 1: Why Finance Runs on Calculus

5 Lessons

What Calculus Actually Is: Change and Accumulation in Markets9 min read

Functions in Finance: Price, Time, Rate, and Payoff Maps10 min read

Limits and Continuity: What Happens as the Interval Shrinks10 min read

Slopes Before Derivatives: Average Return vs Instantaneous Return10 min read

Case Study: Reading a Nifty 50 Intraday Chart as a Function11 min read

Chapter 2: Derivatives as Sensitivities

6 Lessons

The Derivative Defined: From Difference Quotient to Instantaneous Rate11 min read

Power, Sum, and Constant Rules: Differentiating Financial Formulas11 min read

Product, Quotient, and Chain Rules: Differentiating Compound Expressions12 min read

Bond Duration as a Derivative: Price Sensitivity to Yield12 min read

Second Derivatives: Convexity, Gamma, and Curvature of Payoffs12 min read

Case Study: Delta of a Nifty Call Option Read as a Slope13 min read

Chapter 3: Optimization: Finding the Best Point

4 Lessons

Maxima and Minima: Setting the Derivative to Zero11 min read

Marginal Thinking: Cost, Revenue, and Profit at the Margin11 min read

Partial Derivatives: Functions of Several Variables in Portfolios12 min read

Case Study: Minimum Variance Weights for a Two-Stock Nifty Portfolio13 min read

Chapter 4: Integrals as Accumulation

5 Lessons

The Integral Defined: Summing Tiny Pieces Into a Total11 min read

Antiderivatives and the Fundamental Theorem of Calculus11 min read

Integration Rules Finance Uses: Power, Exponential, and Substitution12 min read

Area Under a Distribution: Probabilities and Expected Values12 min read

Case Study: Expected Payoff of a Bank Nifty Option by Integration13 min read

Chapter 5: The Exponential Function and Continuous Compounding

6 Lessons

Discrete to Continuous: Where the Number e Comes From10 min read

The Exponential and Logarithm: Derivatives That Equal Themselves11 min read

Continuous Compounding and Discounting: e^rt and e^-rt11 min read

Log Returns: Why Quants Add Instead of Multiply11 min read

Growth Rates as Derivatives: CAGR, Instantaneous Growth, and Drift11 min read

Case Study: Converting an FD Rate, a G-Sec Yield, and a Nifty CAGR to Continuous Terms12 min read

Chapter 6: Putting It Together: Calculus Inside Pricing Models

5 Lessons

Present Value as an Integral: Continuous Cash Flows and Perpetuities11 min read

Taylor Series: Approximating Price Changes With Duration and Convexity12 min read

A First Look at Black-Scholes: Where Every Piece of Calculus Shows Up13 min read

Numerical Calculus in Python: Finite Differences and Simple Integration12 min read

Case Study: Repricing a 10-Year G-Sec After a 50 bps RBI Move Using Calculus13 min read