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Calculus Derivatives Formulas

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  • Derivative of a constant function
    The derivative of a constant \(c\) is \(0\).
  • Derivative of a sum
    The derivative of \(f(x) + g(x)\) is \(f'(x) + g'(x)\).
  • Product rule
    The derivative of \(f(x)g(x)\) is \(f'(x)g(x) + f(x)g'(x)\).
  • Power rule
    The derivative of \(x^n\) is \(nx^{n-1}\) for any real number \(n\).
  • Constant multiple rule
    The derivative of \(cf(x)\) is \(cf'(x)\) where \(c\) is a constant.
  • Derivative of a difference
    The derivative of \(f(x) - g(x)\) is \(f'(x) - g'(x)\).
  • Quotient rule
    The derivative of \(\frac{f(x)}{g(x)}\) is \(\frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}\).
  • Chain rule
    The derivative of \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\).
  • Derivative of sin x
    The derivative of \(\sin x\) is \(\cos x\).
  • Derivative of cos x
    The derivative of \(\cos x\) is \(-\sin x\).
  • Derivative of tan x
    The derivative of \(\tan x\) is \(\sec^2 x\).
  • Derivative of cot x
    The derivative of \(\cot x\) is \(-\csc^2 x\).
  • Derivative of sec x
    The derivative of \(\sec x\) is \(\sec x \tan x\).
  • Derivative of csc x
    The derivative of \(\csc x\) is \(-\csc x \cot x\).
  • Derivative of sin⁻¹ x
    The derivative of \(\sin^{-1} x\) is \(\frac{1}{\sqrt{1 - x^2}}\).
  • Derivative of cos⁻¹ x
    The derivative of \(\cos^{-1} x\) is \(-\frac{1}{\sqrt{1 - x^2}}\).
  • Derivative of tan⁻¹ x
    The derivative of \(\tan^{-1} x\) is \(\frac{1}{1 + x^2}\).
  • Derivative of sec⁻¹ x
    The derivative of \(\sec^{-1} x\) is \(\frac{1}{|x| \sqrt{x^2 - 1}}\).
  • Derivative of e^x
    The derivative of \(e^x\) is \(e^x\).
  • Derivative of ln |x|
    The derivative of \(\ln |x|\) is \(\frac{1}{x}\).
  • Derivative of b^x
    The derivative of \(b^x\) is \(b^x \ln b\).
  • Derivative of log_b x
    The derivative of \(\log_b x\) is \(\frac{1}{x \ln b}\).