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Chapter 3 | Derivatives
d dx (tan
x ) = sec 2 x
• Derivative of cotangent function d dx (cot x ) =−csc 2 x • Derivative of secant function d dx (sec x ) = sec x tan x • Derivative of cosecant function d dx (csc x ) =−csc x cot x • The chain rule h ′( x ) = f ′ ⎛ ⎝ g ( x ) ⎞ ⎠ g ′( x ) • The power rule for functions h ′( x ) = n ⎛ ⎝ g ( x ) ⎞ ⎠ n −1 g ′( x ) • Inverse function theorem ⎛ ⎝ f −1 ⎞ ⎠ ′( x ) = 1 f ′ ⎛ ⎝ f −1 ( x ) ⎞ ⎠
whenever f ′ ⎛
⎞ ⎠ ≠0 and f ( x ) is differentiable.
⎝ f −1 ( x )
• Power rule with rational exponents d dx ⎛ ⎝ x m / n ⎞ ⎠ = m n x ( m / n )−1 . • Derivative of inverse sine function d dx sin −1 x = 1 1−( x ) 2 • Derivative of inverse cosine function d dx cos −1 x = −1 1−( x ) 2 • Derivative of inverse tangent function d dx tan −1 x = 1 1+( x ) 2 • Derivative of inverse cotangent function d dx cot −1 x = −1 1+( x ) 2 • Derivative of inverse secant function d dx sec −1 x = 1 | x | ( x ) 2 −1 • Derivative of inverse cosecant function d dx csc −1 x = −1 | x | ( x ) 2 −1 • Derivative of the natural exponential function d dx ⎛ ⎝ e g ( x ) ⎞ ⎠ = e g ( x ) g ′( x ) • Derivative of the natural logarithmic function d dx ⎛ ⎝ ln g ( x ) ⎞ ⎠ = 1 g ( x ) g ′( x ) • Derivative of the general exponential function
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