Derivatives and Differentials

Differential equation, abbreviated as \(\Delta\) (Greek symbol Delta) or DIFF, is an equation which contains one or more terms and the derivatives of one variable with respect to the other variable.  The rate at which this moving object changes relative to the other object can be expressed as the derivative of that initial mathematical equation.

Differential Equations  -  These are problems that require the determination of a function satisfying an equation containing one or more derivatives of the unknown function.
Ordinary Differential Equations  -  The unknown function in the equation only depends on one independent variable; as a result only ordinary derivatives appear in the equation.
Partial Differential Equations  -  The unknown function depends on more than one independent variable; as a result partial derivatives appear in the equation.
Order of Differential Equations  - The order of a differential equation (partial or ordinary) is the highest derivative that appears in the equation.
Linearity of Differential Equations  -  A differential equation is linear if the dependant variable and all of its derivatives appear in a linear fashion (they are not multiplied together or squared for example or they are not part of transcendental functions such as sins, cosines, exponentials, etc.).

Difference Between Terms

  • \(\large{ \Delta }\) (Greek) = Meaning: finite, perceptible change - Usage: to find the difference.    
    • means a change in some variable.  Operation:  \( \Delta = initial - final \)  or  \({\large \frac{ \Delta y }{ \Delta x } = \frac{ y_i \;-\; y_f }{ x_i \;-\; x_f}   }\)
  • \(\large{ \delta }\) (Greek) = Meaning: approximation, non exact functions, estimates - Usage: to find an approximation.         
    • means an 'infinitesimal' change like d, but when you don't want to talk about differential forms or derivatives using the other notations.
  • \(\large{ d }\) (Latin) = Meaning: infinitesimal, imperceptible change - Usage: to differentiate.            
    • means an 'infinitesimal' change, or a "differential form."  It's kind of like a limit as \(\large{ \Delta v -> 0 }\), but it is compatible with relative rates or different kinds of limits, so that the notion of a derivative is preserved (in the form dy/dx, for example.)
  • \(\large{ \partial }\) (Cryillic) = Meaning: infinitesimal change - Usage: partial differentiate.               
    • means a "partial" differential.  It's basically the same as d, except it also tells you that there are other related variables that are being held constant. In other words, it is never a complete picture.  It's typically used in partial derivatives -- derivatives that are only in one dimension of a larger dimensional space.
  • \(\large{ \delta }\), \(\large{ d }\), and \(\large{ \partial }\) are not used interchangeably.

Derivative

A derivative is a math expression of a rate of change.  Any rate of change of a moving object related to the other object can be expressed as a derivative.      

A derivative is a rate of change, which is the slope of a graph in geometric terms.  \({\large slope = \frac{ change\; in \;y }{ change\; in \; x } }\)

The mark \('\) means derivative of, and \(f\) and \(g\) are functions.

The slope of a constant value is always 0.

 

Derivative Rules
Rules
 Function Derivative
Constant Function Rule \(\large{ \frac{d}{dx}\;a = 0 }\)  (where a = any constant)
Scalar Multiple Rule \(\large{ \frac{d}{dx}\left(au \right) = a\;\frac{du}{dx}  }\)  (where a = any constant)
Sum Rule \(\large{ \frac{d}{dx}\left(u+v \right) = \frac{du}{dx} + \frac{dv}{dx}  }\)
Difference Rule \(\large{ \frac{d}{dx}\left(u-v \right) = \frac{du}{dx} - \frac{dv}{dx}  }\)
Power Rule \(\large{ \frac{d}{dx} u^n = nu^{n-1} \frac{du}{dx}  }\)
Product Rule \(\large{ \frac{d}{dx}\left(uv \right) = u'v+uv' }\)
Quotient Rule \(\large{ \frac{d}{dx}\left(\frac{u}{v} \right) = \frac{u'v-uv'}{v^2} }\)
Reciprocal Rule \(\large{ \frac{d}{dx}\left(\frac{1}{v} \right) = \frac{v'}{v^2} }\)
Chane Rule \(\large{ \frac{d}{dx} f \left( g\left( x \right) \right)  = f' \left( g\left( x \right) \right) g'\left( x \right) }\)  or  \(\large{ \frac{dy}{dx} = \frac{dy}{du}\;\frac{du}{dx} }\)
Trig Function Rule

\(\large{ \frac{d}{dx} \; sin\;u  = cos\;u \;\frac{du}{dx} }\)                      \(\large{ \frac{d}{dx} \; cos\;u  = sin\;u \;\frac{du}{dx} }\)

\(\large{ \frac{d}{dx} \; tan\;u  = sec^2\;u \;\frac{du}{dx} }\)                   \(\large{ \frac{d}{dx} \; cot\;u  = csc^2\;u \;\frac{du}{dx} }\)

\(\large{ \frac{d}{dx} \; sec\;u  = sec\;u \;tan\;u \;\frac{du}{dx} }\)          \(\large{ \frac{d}{dx} \; csc\;u  = csc\;u \;cot\;u \;\frac{du}{dx} }\)
Inverse Trig Function Rule

\(\large{ \frac{d}{dx} \; sin^{-1}\;u  = \frac{1}{\sqrt{1\;-\;u^2 } } \;\frac{du}{dx} }\)                      \(\large{ \frac{d}{dx} \; cos^{-1}\;u  = -\; \frac{1}{\sqrt{1\;-\;u^2 } } \;\frac{du}{dx} }\)

\(\large{ \frac{d}{dx} \; tan^{-1}\;u  = \frac{1}{1\;+\;u^2 }  \;\frac{du}{dx} }\)                   \(\large{ \frac{d}{dx} \; cot^{-1}\;u  = -\; \frac{1}{1\;+\;u^2 } \;\frac{du}{dx} }\)

\(\large{ \frac{d}{dx} \; sec^{-1}\;u  = \frac{1}{\left\vert u \right\vert\;\sqrt{u^2\;-\; 1 }  } \;\frac{du}{dx} }\)          \(\large{ \frac{d}{dx} \; csc^{-1}\;u  = -\; \frac{1}{\left\vert u \right\vert\;\sqrt{u^2\;-\; 1 }  } \;\frac{du}{dx} }\)
Expotential Function Rule   \(\large{ \frac{d}{dx} \left( \epsilon^{\alpha} \right) = \epsilon^{\alpha} \; \frac{du}{dx} }\)  and  \(\large{ \frac{d}{dx} \left( a^{\alpha} \right)  = a^{\alpha} \left( ln\;a \right) \; \frac{du}{dx}  }\)
Log Function Rule  \(\large{ \frac{d}{dx} \left( ln\;u \right) = \frac{1}{u} \; \frac{du}{dx} }\)  and  \(\large{ \frac{d}{dx} \left( log_{\lambda} \;u \right)  = \frac{1}{ \left( ln\;a \right) u } \; \frac{du}{dx}  }\)
Inverse Function Rule \(\large{ \frac{d}{dx} f^{-1} \left( x \right) = \frac{ 1 }{ f' \left( f^{-1} \left( x \right) \right) }  }\)

 

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