Derivatives Formulas and Notes

Contents

Definition of a Derivative Properties of a Derivative Power Rule Product Rule Quotient Rule Chain Rule Exponential and Logarithm Functions Trigonometric Functions Inverse Trigonometric Functions Hyperbolic Functions Common Derivatives

Definition of a Derivative

The derivative of f (x) with respect to x is the function f ′(x) and is defined as, \displaystyle f'(x) = \lim_{h \to 0} { \frac{ {f(x+h)-f(x)} }{h} }.

Example
Find the derivative of \displaystyle f(x) = x^{2}

\displaystyle f'(x) = \lim_{h \to 0} { \frac{ {f(x+h)-f(x)} }{h} }
\displaystyle f'(x) = \lim_{h \to 0} { \frac{ {(x+h)^{2}-x^{2}} }{h} }
\displaystyle f'(x) = \lim_{h \to 0} { \frac{ {x^{2}+2xh+h^{2}-x^{2}} }{h} }
\displaystyle f'(x) = \lim_{h \to 0} { \frac{ {2xh+h^{2}} }{h} }
\displaystyle f'(x) = \lim_{h \to 0} { \frac{ {h(2x+h)} }{h} }
\displaystyle f'(x) = \lim_{h \to 0} { {2x+h} }
\displaystyle f'(x) = { {2x+(0) } }
\displaystyle f'(x) = 2x

Properties of Derivatives

\displaystyle \frac{d}{dx}(c)=0

\displaystyle \frac{d}{dx}(cf(x)) = cf'(x)

\displaystyle \frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)

Power Rule

If \displaystyle f(x) = x^{n} then \displaystyle f(x) = nx^{n-1} .

Example
Find the derivative of \displaystyle f(x) = x^{4}
\displaystyle f'(x) = 4 \cdot x^{4-1}
\displaystyle f'(x) = 4x^{3}

Find the derivative of \displaystyle f'(x) = 4x^{6}

\displaystyle f'(x) = 6 \cdot 4x^{6-1}
\displaystyle f'(x) = 24x^{5}

Product Rule

\displaystyle \frac{d}{dx}(f(x)g(x)) = {f'(x)g(x)+f(x)g'(x)}

Example
Find the derivative of \displaystyle (x^{4})(x^2-4x)
Let \displaystyle f(x) = x^{4}, g(x) = x^2-4x
\displaystyle f'(x) = 4x^{3}, g'(x) = 2x-4
\displaystyle \frac{d}{dx}(f(x)g(x)) = {f'(x)g(x)+f(x)g'(x)}
\displaystyle = {(4x^{3})(x^2-4x)+(x^{4})(2x-4)}
\displaystyle = { 4x^{5} - 16x^{4}+ 2x^{5}-4x^{4} }
\displaystyle = { 6x^{5} - 20x^{4} }

Quotient Rule

\displaystyle \frac{d}{dx} \left( \frac{ {f(x)} }{ {g(x)} } \right) = \frac{f'(x)g(x)-f(x)g'(x)}{ {g(x)}^2}

Example
Find the derivative of \displaystyle \frac{x^{2}}{x^{4}-3}
Let \displaystyle f(x) = x^{2}, g(x) = x^{4}-3
\displaystyle f'(x) = 2x, g'(x) = 4x^{3}
\displaystyle \frac{d}{dx} \left( \frac{ {f(x)} }{ {g(x)} } \right) = \frac{f'(x)g(x)-f(x)g'(x)}{ {g(x)}^2}

\displaystyle = \frac{ (2x)(x^{4}-3)-(x^{2})(4x^{3})}{ {(x^{4}-3)}^2}

\displaystyle = \frac{ 2x^{5} - 6x -4x^{5} }{ {(x^{4}-3)}^2}

\displaystyle = \frac{ -2x^{5} - 6x }{ {(x^{4}-3)}^2} \displaystyle = \frac{ -2x( x^{4}+3) }{ {(x^{4}-3)}^2}

Chain Rule

Suppose that we have two functions f(x) and g(x) and they are both differentiable.
If y=f(u) and u=g(x) then the derivative of y is, \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}

Example
Find the derivative of \displaystyle y = ({x^{2}-5x})^{2}
\displaystyle f(u) = (u)^{2} , \displaystyle u = {x^{2}-5x}
\displaystyle \frac{dy}{du} = 2(u) , \displaystyle \frac{du}{dx} = 2x-5
\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}
= 2(u) \cdot (2x-5)
Remember that \displaystyle u = {x^{2}-5x} .
2(u) \cdot (2x-5) = 2({x^{2}-5x}) \cdot (2x-5)
= (2x^{2}-10x)(2x-5)

Exponential and Logarithm Functions

\displaystyle \frac{d}{dx}(a^{x}) = a^{x} \ln(a)

Example
Find the derivative of \displaystyle f(x) = 5^{x}
\displaystyle f'(x) = 5^{x} \ln(5)
Example
Find the derivative of \displaystyle f(x) = e^{x}
\displaystyle f'(x) = e^{x} ( \ln(e))
\displaystyle f'(x) = e^{x} (1)
\displaystyle f'(x) = e^{x}

\displaystyle \frac{d}{dx}( \ln(x) ) = \frac{1}{x}

\displaystyle \frac{d}{dx}( \log_{a}{(x)} ) = \frac{1}{ x \ln(a)}

Example
Find the derivative of \displaystyle f(x) = \log_{6}{(x)}
\displaystyle f'(x) = \frac{1}{ x \ln(6)}

Trigonometric Functions

\displaystyle \frac{d}{dx}(\sin(x)) = \cos(x)

\displaystyle \frac{d}{dx}(\cos(x)) = -\sin(x)

\displaystyle \frac{d}{dx}(\tan(x)) = \sec^2(x)

\displaystyle \frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)

\displaystyle \frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)

\displaystyle \frac{d}{dx}(\cot(x)) = -\csc^2(x)

Inverse Trigonometric Functions

\displaystyle \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{ { \sqrt{1-x^2} } }

\displaystyle \frac{d}{dx}(\cos^{-1}(x)) = \frac{-1}{ { \sqrt{1-x^2} } }

\displaystyle \frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{ { {1+x^2} } }

\displaystyle \frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{ { |x| \sqrt{x^2-1} } }

\displaystyle \frac{d}{dx}(\csc^{-1}(x)) = - \frac{1}{ { |x| \sqrt{x^2-1} } }

\displaystyle \frac{d}{dx}(\cot^{-1}(x)) = - \frac{1}{ { {1+x^2} } }

Hyperbolic Functions

\displaystyle \frac{d}{dx}(\sinh(x)) = \cosh(x)

\displaystyle \frac{d}{dx}(\cosh(x)) = \sinh(x)

\displaystyle \frac{d}{dx}(\tanh(x)) = sech^2(x)

\displaystyle \frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{x^{2} + 1}}

\displaystyle \frac{d}{dx}(\cosh^{-1}(x)) = \frac{1}{\sqrt{x^{2} - 1}}

\displaystyle \frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{ 1 - x^{2} }

\displaystyle \frac{d}{dx}(sech(x)) = -sech(x)\tanh(x)

\displaystyle \frac{d}{dx}(csch(x)) = -csch(x) coth(x)

\displaystyle \frac{d}{dx}(coth(x)) = -csch^2(x)

Common Derivatives

\displaystyle \frac{d}{dx}( x^{n} ) = nx^{n-1}