07Hproductrule

toc = The Product Rule =


 * Product ** means "the result of multiplying" so use this when two functions have been multiplied together.

If u and v are functions of x {ie u(x) and v(x)} then:
 * If //f (x) = u × v//,
 * then //f '(x) = u// × //v' + u'// × //v//

math \\ . \qquad \text{If } y = u \times v \qquad. \\ . \\ . \qquad \qquad \dfrac{dy}{dx}=u\times\dfrac{dv}{dx}+v\times\dfrac{du}{dx} \qquad. math
 * OR**

** Example 1 **
Find the derivative of each of the following with respect to //x//: (a) //y = (2x 2 + 1)(5x 3 +16)// (b) //y = x 3 (3x – 5) 4 // (c) //y = sin (x) log e (x)//

... ... //y = (2x 2 + 1)(5x 3 +16)//
 * __Solution:__**
 * (a) **

... ... //u = (2x 2 + 1)// ... and ... //v = (5x 3 +16)//

math .\qquad\dfrac{du}{dx}=4x\qquad \qquad \dfrac{dv}{dx}=15x^2 \qquad. math

math .\qquad\dfrac{dy}{dx}=u\times\dfrac{dv}{dx}+v\times\dfrac{du}{dx} \qquad. math

math .\qquad\quad=(2x^2+1)\times15x^2+(5x^3+16)\times4x \qquad. math

math .\qquad\quad=15x^2(2x^2+1)+4x(5x^3+16) \qquad. \\ . \\ . \qquad \quad = 30x^4 + 15x^2 + 20x^4 + 64x \qquad. \\ . \\ . \qquad \quad = 50x^4 + 15x^2 + 64x math

... ... //y = x 3 (3x – 5) 4 //
 * (b) **

... ... ... //u = x 3 // ... and ... //v = (3x – 5) 4 //

math .\qquad\dfrac{du}{dx}=3x^2\qquad \qquad \dfrac{dv}{dx}=3\times4\times(3x-5)^3 \qquad. math

math .\qquad\dfrac{dy}{dx}=u\times\dfrac{dv}{dx}+v\times\dfrac{du}{dx} \qquad. math

math .\qquad\quad=x^3\times12(3x-5)^3+(3x-5)^4\times3x^2 \qquad. \\ . \\ .\qquad\quad=12x^3(3x-5)^3+3x^2(3x-5)^4 \qquad. math

... ... //y = sin (x) log e (x)//
 * (c) **

... ... ... //u = sin (x)// ... and ... //v = log e (x)//

math .\qquad\dfrac{du}{dx}=cos(x)\qquad\dfrac{dv}{dx}=\dfrac{1}{x} \qquad. math

math .\qquad\dfrac{dy}{dx}=u\times\dfrac{dv}{dx}+v\times\dfrac{du}{dx} \qquad. \\ . \\ .\qquad\quad=sin(x)\times\dfrac{1}{x}+log_e(x)\times{cos(x)} \qquad. \\ . \\ .\qquad\quad=\dfrac{sin(x)}{x}+cos(x)log_e(x) math Go to top of page flat

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