Calculus Derivative Rules

Calculus Derivative Rules

Page 1

Power Rule

• Formula: ( \frac{d}{dx} x^n = n \cdot x^{n-1} )

Product Rule

• Formula: ( \frac{d}{dx} f(x) \cdot g(x) = f(x) \cdot g'(x) + g(x) \cdot f'(x) )

Quotient Rule

• Formula: ( \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{(g(x))^2} )

Section 3.5 (p. 135)

Trigonometric Derivatives

• Sine Function:

  • ( y = \sin x )

  • ( y' = \cos x )

• Cosine Function:

  • ( y = \cos x )

  • ( y' = -\sin x )

• Tangent Function:

  • ( y = \tan x )

  • ( y' = \sec^2 x )

• Cosecant Function:

  • ( y = \csc x )

  • ( y' = -\csc x \cdot \cot x )

• Secant Function:

  • ( y = \sec x )

  • ( y' = \sec x \cdot \tan x )

• Cotangent Function:

  • ( y = \cot x )

  • ( y' = -\csc^2 x )

Section 3.9 (p. 164)

Exponential and Logarithmic Derivatives

• Exponential Function:

  • ( y = e^x )

  • ( y' = e^x )

• Exponential with Base 5:

  • ( y = 5^x )

  • ( y' = \ln 5 \cdot 5^x )

• Natural Logarithm:

  • ( y = \ln x )

  • ( y' = \frac{1}{x} )

• Logarithm with Base ( a ):

  • ( y = \log_a u )

  • ( y' = \frac{1}{u \cdot \ln(a)} )

Section 3.8 (p. 159)

Inverse Trigonometric Derivatives

• Inverse Sine:

  • ( y = \sin^{-1}(u) )

  • ( y' = \frac{1}{\sqrt{1-u^2}} )

• Inverse Cosine:

  • ( y = \cos^{-1}(u) )

  • ( y' = -\frac{1}{\sqrt{1-u^2}} )

• Inverse Tangent:

  • ( y = \tan^{-1}(u) )

  • ( y' = \frac{1}{1+u^2} )

• Inverse Cosecant:

  • ( y = \csc^{-1}(u) )

  • ( y' = -\frac{1}{|u| \sqrt{u^2-1}} )

• Inverse Secant:

  • ( y = \sec^{-1}(u) )

  • ( y' = \frac{1}{|u| \sqrt{u^2-1}} )

• Inverse Cotangent:

  • ( y = \cot^{-1}(u) )

  • ( y' = -\frac{1}{1+u^2} )

Section 3.6 (p. 142)

Chain Rule

• Formula:

  • If ( y = f(g(x)) ), then ( y' = f'(g(x)) \cdot g'(x) )

• Example:

  • For ( y = (5x + 7)^2 ):

    • ( y' = 2(5x + 7)(5) = 10(5x + 7) = 50x + 70 )