Differential Calculus Explained Simply and Clearly

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of rates of change of quantities. It focuses on how a function changes as its input changes and is widely used in mathematics, physics, and engineering.

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Key Concepts

• Function: A relation between two variables
• Derivative: Measures the rate of change of a function
• Differentiation: Process of finding derivatives
• Slope of a curve: Represents the derivative at a point

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Basic Rules of Differentiation

1. Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$dxd​xn=nxn−1
2. Sum and Difference Rule: $\frac{d}{dx} [f(x) ± g(x)] = f'(x) ± g'(x)$dxd​[f(x)±g(x)]=f′(x)±g′(x)
3. Product Rule: $\frac{d}{dx}[u.v] = u’v + uv’$dxd​[u.v]=u′v+uv′
4. Quotient Rule: $\frac{d}{dx}\frac{u}{v} = \frac{u’v – uv’}{v^2}$dxd​vu​=v2u′v−uv′​
5. Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$dxd​f(g(x))=f′(g(x))g′(x)

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Applications of Differential Calculus

• Finding slopes and tangents of curves
• Calculating velocity and acceleration in physics
• Optimization problems in economics and engineering
• Analyzing rates of change in real-world systems

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Conclusion

Differential calculus is a fundamental mathematical tool to study how quantities change. Mastering derivatives and their rules is essential for solving problems in mathematics, science, and engineering.