Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes. The basic rule of differentiation states that if we have a differentiable function f(x), then its derivative f'(x) gives us the slope of the tangent line to the graph of f(x) at any point x.

There are several different ways to find derivatives, including:

1. Limit definition: f'(x) = lim (h->0)[f(x+h)-f(x)]/h
2. Power rule: If f(x)=x^n, then f'(x)=nx^(n-1)
3. Product rule: If f(x)=u(x)v(x), then f'(x)=u'(x)v(x)+u(x)v'(x)
4. Quotient rule: If f(x)=u(x)/v(x), then f'(x)=[v(x)*u'(x)-u(x)*v'(x)]/(v(x)^2)
5. Chain rule: If f(g(x))=u(g(x)), then f'(x)=u'(g(x))*g'(x)
6. Sum and difference rules: For f(x)=u(x)+v(x) or f(x)=u(x)-v(x), we have f'(x)=u'(x)+v'(x) and f'(x)=u'(x)-v'(x), respectively.

These rules allow us to differentiate a wide variety of functions, leading to many important applications in physics, engineering, economics, and other fields.
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence