In the context of calculus, what are the basic differentiation rules?

In calculus, differentiation is a fundamental concept that involves finding the rate of change of a function with respect to its independent variable. There are several basic rules or techniques that help simplify the process of differentiating various types of functions. Here are the basic differentiation rules:

1. 1. Constant Rule:

   d/dx(c) = 0

   2. Power Rule:

   d/dx(x^n) = n * x^(n-1)

   3. Sum/Difference Rule:

   d/dx(f(x) + g(x)) = d/dx(f(x)) + d/dx(g(x)) d/dx(f(x) - g(x)) = d/dx(f(x)) - d/dx(g(x))

   4. Product Rule:

   d/dx(f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)

   5. Quotient Rule:

   d/dx(f(x) / g(x)) = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2

   6. Chain Rule:

   d/dx(h(f(x))) = h'(f(x)) * f'(x)

   7. Trigonometric Function Rules: (Various rules for derivatives of trigonometric functions)

   8. Exponential and Logarithmic Function Rules: (Various rules for derivatives of exponential and logarithmic functions)

   9. Implicit Differentiation: (Technique for finding dy/dx when you have an equation involving x and y)

These basic rules form the foundation of differentiation and are used to find the derivatives of a wide variety of functions. By applying these rules appropriately, you can compute derivatives of more complex functions and solve problems in areas such as physics, engineering, economics, and more.