Derivative Rules

Derivative Rules

Power Rule

• If f(x) = x^n, then f'(x) = nx^(n-1)
• Example: If f(x) = x^2, then f'(x) = 2x

Product Rule

• If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
• Example: If f(x) = x^2 sin(x), then f'(x) = 2x sin(x) + x^2 cos(x)

Quotient Rule

• If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
• Example: If f(x) = sin(x) / x, then f'(x) = (cos(x) \* x - sin(x)) / x^2

Chain Rule

• If f(x) = g(h(x)), then f'(x) = g'(h(x)) \* h'(x)
• Example: If f(x) = sin(x^2), then f'(x) = 2x cos(x^2)

Sum and Difference Rule

• If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x)
• Example: If f(x) = x^2 + sin(x), then f'(x) = 2x + cos(x)

These rules can be applied to find the derivative of various functions.

Derivative Rules

Power Rule

• The derivative of f(x) = x^n is f'(x) = nx^(n-1)
• Applies to functions of the form f(x) = x raised to a power n

Product Rule

• The derivative of f(x) = u(x)v(x) is f'(x) = u'(x)v(x) + u(x)v'(x)
• Applies to functions that are the product of two functions u(x) and v(x)

Quotient Rule

• The derivative of f(x) = u(x)/v(x) is f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
• Applies to functions that are the quotient of two functions u(x) and v(x)

Chain Rule

• The derivative of f(x) = g(h(x)) is f'(x) = g'(h(x)) * h'(x)
• Applies to composite functions, where a function g is composed with h(x)

Sum and Difference Rule

• The derivative of f(x) = u(x) ± v(x) is f'(x) = u'(x) ± v'(x)
• Applies to functions that are the sum or difference of two functions u(x) and v(x)

Quiz on the rules of derivatives in calculus, including the power rule, product rule, quotient rule, and chain rule.