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Difference between revisions of "Differentiation Rules"

(Created page with 'Differentiation Rules are rules used to compute the derivative of a function in calculus. ==Basic Rules== '''Derivative of a Constant:''' If <math>y=c</math> and c is any consta…')
 
(→‎Derivatives of Trig Functions)
 
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Differentiation Rules are rules used to compute the derivative of a function in calculus.
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'''Differentiation rules''' are rules (actually, theorems) used to compute the [[derivative]] of a [[function]] in calculus.  In what follows, all functions are assumed to be differentiable.
  
 
==Basic Rules==
 
==Basic Rules==
 
'''Derivative of a Constant:'''
 
'''Derivative of a Constant:'''
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If <math>y=c</math> and c is any constant then <math>\frac{dy}{dx} = 0</math>
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If <math>y(x)=c</math> is a [[constant]] function then <math>\frac{dy}{dx} = 0</math>.
  
 
'''Sum Rule:'''
 
'''Sum Rule:'''
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If <math>y(x) = u(x)+v(x)</math> then <math>\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}</math>
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If <math>y(x) = u(x)+v(x)</math> then <math>\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}</math>.
  
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'''Product Rule:'''
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[[Product Rule]]:
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If <math>y(x) = u(x) * v(x)</math> then <math>\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}</math>
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If <math>y(x) = u(x) \cdot v(x)</math> then <math>\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}</math>.
  
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'''Quotient Rule:'''
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[[Quotient Rule]]:
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If <math>y(X) = \frac{u(x)}{v(x)}</math> then <math>\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}</math>
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If <math>y(x) = \frac{u(x)}{v(x)}</math> then <math>\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}</math>.
  
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'''Chain Rule:'''
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[[Chain Rule]]:
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If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}*\frac{dv}{dx}</math>
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If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}</math>.
  
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'''power Rule:'''
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'''Power Rule:'''
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If <math>y(x) = u(x^n)</math> then <math>\frac{dy}{dx} = n(u(x))^{n-1} * \frac{du}{dx}</math>
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If <math>y(x) = (u(x))^n</math> then <math>\frac{dy}{dx} = n(u(x))^{n-1} \cdot \frac{du}{dx}</math>.  For [[integer]] <math>n</math> this is just a consequence of the product and quotient rules and [[induction]], but it can also be proven for all [[real number]]s <math>n</math>, e.g. by using the extended [[Binomial Theorem]].
  
 
==Derivatives of Trig Functions==
 
==Derivatives of Trig Functions==
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'''Derivative of Sine'''
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If <math>y(x) = \sin x</math>, then <math>\frac{dy}{dx} = \cos x</math>.
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'''Derivative of Cosine'''
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If <math>y(x) = \cos x</math>, then <math>\frac{dy}{dx} = -\sin x</math>.
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'''Derivative of Tangent'''
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If <math>y(x) = \tan x</math>, then <math>\frac{dy}{dx} = \sec^2 x</math>. Note that this follows from the Quotient Rule.
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'''Derivative of Cosec'''
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If <math>y(x) = \csc x</math>, then <math>\frac{dy}{dx} = -\csc(x)\cot(x)</math>.

Latest revision as of 09:50, 4 June 2024

Differentiation rules are rules (actually, theorems) used to compute the derivative of a function in calculus. In what follows, all functions are assumed to be differentiable.

Basic Rules

Derivative of a Constant: If $y(x)=c$ is a constant function then $\frac{dy}{dx} = 0$.

Sum Rule: If $y(x) = u(x)+v(x)$ then $\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}$.

Product Rule: If $y(x) = u(x) \cdot v(x)$ then $\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}$.

Quotient Rule: If $y(x) = \frac{u(x)}{v(x)}$ then $\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}$.

Chain Rule: If $y(x) = u(v(x))$ then $\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}$.

Power Rule: If $y(x) = (u(x))^n$ then $\frac{dy}{dx} = n(u(x))^{n-1} \cdot \frac{du}{dx}$. For integer $n$ this is just a consequence of the product and quotient rules and induction, but it can also be proven for all real numbers $n$, e.g. by using the extended Binomial Theorem.

Derivatives of Trig Functions

Derivative of Sine If $y(x) = \sin x$, then $\frac{dy}{dx} = \cos x$.

Derivative of Cosine If $y(x) = \cos x$, then $\frac{dy}{dx} = -\sin x$.

Derivative of Tangent If $y(x) = \tan x$, then $\frac{dy}{dx} = \sec^2 x$. Note that this follows from the Quotient Rule.

Derivative of Cosec If $y(x) = \csc x$, then $\frac{dy}{dx} = -\csc(x)\cot(x)$.

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