Partial derivative and Chain rule: Difference between pages

Latest revision as of 04:58, 22 July 2016

Template:Multiple issues The chain rule is a way of finding the derivative of a function. It is used where the function is in another function. This is called a composite function.

If F(x) equals two functions that we can take a derivative of, such as:

[math]\displaystyle{ F(x)=f(g(x)) }[/math]

then the derivative, F prime, is

[math]\displaystyle{ F'(x)=f'(g(x))g'(x) }[/math]

Steps

1. Find the derivative of the outside function (all of it at once).

2. Find the derivative of the inside function (the bit between the brackets).

3. Multiply the answer from the first step by the answer from the second step. This is basically the last step in solving for the derivative of a function.

Example;

[math]\displaystyle{ F(x)=(x^2+5)^3 }[/math]

[math]\displaystyle{ F(x)=3(x^2+5)^2 }[/math]

[math]\displaystyle{ F'(x)=3(x^2+5)^2(2x) }[/math]

[math]\displaystyle{ F'(x)=6x(x^2+5)^2 }[/math]

In this example, the cubed sign (³) is the outside function and [math]\displaystyle{ x^2+5 }[/math] is the inside function. The derivative of the outside function would be [math]\displaystyle{ 3x^2 }[/math], where the inside function is plugged in for x. The derivative of the inside function would be 2x, which is multiplied by [math]\displaystyle{ 3(x^2+5)^2 }[/math] to get [math]\displaystyle{ 6x(x^2+5)^2 }[/math].

Template:Stub

@@ Line 1: / Line 1: @@
-In [[calculus]], an advanced branch of [[mathematics]], the '''partial derivative''' of a [[Function (mathematics)|function]] is basically the derivative of one named variable, and the unnamed variable of the function is held constant. In other words, the partial derivative takes the [[derivative]] of certain indicated variables of a function and doesn't differentiate the other variable(s). The notation
+{{Multiple issues|wikify=October 2012}}
+The '''chain rule''' is a way of finding the derivative of a function. It is used where the function is in another function. This is called a composite function.
-<math>\frac{\partial f}{\partial x}</math>
+If F(x) equals two functions that we can take a derivative of, such as:
+:<math>F(x)=f(g(x))</math>
-is usually used, although other notations are valid. Usually, although not in all cases, the partial derivative is taken in a multivariable function, (i.e., the function has three or more variables, whether independent or dependent variables).
+then the derivative, F prime, is
+:<math>F'(x)=f'(g(x))g'(x)</math>
-== Examples ==
+==Steps==
-If we have a function <math>f(x, y)=x^2+y</math>, then there are several partial derivatives of ''f(x, y)'' that are all equally valid. For example,
+'''1.''' Find the derivative of the outside function (all of it at once).
-:<math>\frac{\partial }{\partial y}[f(x, y)]=1</math>
+'''2.''' Find the derivative of the inside function (the bit between the brackets).
-Or, we can do the following
+'''3.''' Multiply the answer from the first step by the answer from the second step. This is basically the last step in solving for the derivative of a function.
-:<math>\frac{\partial}{\partial x}[f(x, y)]=2x</math>
+;Example;
-==Related pages==
+:<math>F(x)=(x^2+5)^3</math>
-*[[Derivative (mathematics)]]
+:<math>F(x)=3(x^2+5)^2</math>
+:<math>F'(x)=3(x^2+5)^2(2x)</math>
+:<math>F'(x)=6x(x^2+5)^2</math>
-*[[Calculus]]
-*[[Difference quotient]]
+In this example, the cubed sign (<sup>3</sup>) is the outside function and <math>x^2+5</math> is the inside function. The derivative of the outside function would be <math>3x^2</math>, where the inside function is plugged in for x. The derivative of the inside function would be 2x, which is multiplied by <math>3(x^2+5)^2</math> to get <math>6x(x^2+5)^2</math>.
-[[Category:Calculus]]
+{{stub}}
+[[Category:Mathematics]]

Partial derivative and Chain rule: Difference between pages

Latest revision as of 04:58, 22 July 2016

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