Differentiation Methods

 

Remember that the gradient of a straight line y=mx+b is m. The tangent to the line is the line itself, so the gradient of tangent is m everywhere along the line.

 

So if y=mx, \cfrac{dy}{dx}=m.

 

\cfrac { d }{ dx } (kx)=k

For a horizontal line in the form y=k, the gradient is zero.

So if y=k, \cfrac{dy}{dx}=0.

\cfrac { d }{ dx } (k)=0

In short, we could say that:

\cfrac { d }{ dx } ({ x }^{ n })=n{ x }^{ n-1 }

A more general way of writing the rule is:

\cfrac { d }{ dx } ({ kx }^{ n })=kn{ x }^{ n-1 }

OR

\cfrac { d }{ dx } (kf\left( x \right) )=kf^{ ' }\left( x \right)

Also, if there are several terms in an expression, we differentiate each one separately. We can write this as a rule:

\cfrac { d }{ dx } (f\left( x \right) +g\left( x \right) )=f^{ ' }\left( x \right) +g^{ ' }\left( x \right)

 

Examples:

 

Differentiate each function:

1. 7x

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\cfrac { d }{ dx } (7x)=7

 

2. y=4{x}^{7}

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\cfrac { dy }{ dx } =4\times 7{ x }^{ 6 }=28{ x }^{ 6 }

 

3. f(x)={ x }^{ 4 }-{ x }^{ 3 }+5

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f'(x)=4{ x }^{ 3 }-3{ x }^{ 2 }+0\\ f'(x)=4{ x }^{ 3 }-3{ x }^{ 2 }

 

4. If f(x)=2{ x }^{ 5 }-7{ x }^{ 3 }+5x-4, evaluate f'(-1)

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f'(x)=10{ x }^{ 4 }-21{ x }^{ 2 }+5\\ f'(-1)=10{ (-1) }^{ 4 }-21{ (-1) }^{ 2 }+5\\ f'(-1)=-6

 

5. Differentiate S=2\pi {r}^{2}+2\pi rh with respect to r.

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In this case, we are differentiating with r as the variable, so \pi and h are constants.

\cfrac { dS }{ dr } =2\pi (2r)+2\pi h\\ \\ \cfrac { dS }{ dr } =4\pi r+2\pi h

 

Composite Function Rule

 

composite function is a function composed of two or more other functions. For example, {(3x^2 - 4)}^5 is made up of a function u^5 where u=3x^2 - 4.

To differentiate a composite function, we need to use the result:

\cfrac { dy }{ dx } =\cfrac { dy }{ du } \times \cfrac { du }{ dx }

 

Examples

1. Differentiate {(5x+4)}^7

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2. Differentiate \sqrt { 3-x }

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The derivative of a composite function is the product of two derivatives. One is the derivative of the function inside the brackets. The other is the derivative of the whole function.

\cfrac { d }{ dx } [f\left( x \right) ]^{ n }=f^{ ' }\left( x \right) n[f\left( x \right) ]^{ n-1 }

 

 

Examples:

 

Differentiate:

 

1. { (8{ x }^{ 3 }-1) }^{ 5 }

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\cfrac { dy }{ dx } =24{ x }^{ 2 }\cdot 5{ (8{ x }^{ 3 }-1) }^{ 4 }\\ \\ \cfrac { dy }{ dx } =120{ x }^{ 2 }{ (8{ x }^{ 3 }-1) }^{ 4 }

 

2. \cfrac { 1 }{ { (6x+1) }^{ 2 } }

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y={ (6x+1) }^{ -2 }\\ y'=6\times -2{ (6x+1) }^{ -3 }\\ y'=-12{ (6x+1) }^{ -3 }\\ \\ y'=-\cfrac { 12 }{ { (6x+1) }^{ 3 } }

 

 

Product and Quotient Rule

 

Differentiating the product of two functions y=uv gives the result:

\cfrac { dy }{ dx } =u\cfrac { dv }{ dx } +v\cfrac { du }{ dx }

If y=uv, y'=u'v+v'u

Differentiating the quotient of two function y=\cfrac{u}{v} gives the result:

\cfrac { dy }{ dx } =\cfrac { v\cfrac { du }{ dx } -u\cfrac { dv }{ dx }  }{ { v }^{ 2 } }

If y=\cfrac{u}{v}, y'=\cfrac { vu'-uv' }{ v^2 }

 

Examples

 

1. Differentiate 2{ x }^{ 5 }{ (5x+3) }^{ 3 }

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u=2{ x }^{ 5 }\quad and\quad v={ (5x+3 })^{ 3 }\\ u'=10{ x }^{ 4 }\quad and\quad v'=5\cdot 3{ (5x+3 })^{ 2 } \\ y'=u'v+v'u\\ y'=10{ x }^{ 4 }{ (5x+3) }^{ 3 }+5\cdot 3{ (5x+3) }^{ 2 }\cdot 2{ x }^{ 5 }\\ y'=10{ x }^{ 4 }{ (5x+3) }^{ 3 }+30{ x }^{ 5 }{ (5x+3) }^{ 2 }\\ y'=10{ x }^{ 4 }{ (5x+3) }^{ 2 }[(5x+3)+3x]\\ y'=10{ x }^{ 4 }{ (5x+3) }^{ 2 }(8x+3))

 

2. Differentiate (3x-4) \sqrt(5-2x)

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3. Differentiate \cfrac{3x-5}{5x+2}

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4. Differentiate \cfrac{4x^3-5x+2}{x^3-1}

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