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GRADE 12 – Unit 2: Introduction to Calculus
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Derivatives of Combinations of Functions

 

  • Sum or Difference of Functions:

If f(x)f(x) and g(x)g(x) are differentiable, then:

(f+g)(x)=f(x)+g(x)(f + g)'(x) = f'(x) + g'(x)
(fg)(x)=f(x)g(x)(f – g)'(x) = f'(x) – g'(x)

This means if you want to find the rate of change of two functions added or subtracted, you can find the derivative of each and then add or subtract them.

 

  • Scalar Multiplication:

If kk is a constant and f(x)f(x) is differentiable:

(kf)(x)=kf(x)(kf)'(x) = k f'(x)

This means if you multiply a function by a number, you can find the derivative by multiplying that number by the derivative of the function.

 

  • Product Rule:

If f(x)f(x) and g(x)g(x) are differentiable, then:

(fg)(x)=f(x)g(x)+f(x)g(x)(fg)'(x) = f'(x)g(x) + f(x)g'(x)

This is called the product rule, which is used when two functions are multiplied together.

 

  • Quotient Rule:

If f(x)f(x) and g(x)g(x) are differentiable, then the derivative of the quotient is:

(fg)(x)=f(x)g(x)f(x)g(x)g(x)2left( frac{f}{g} right)'(x) = frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2}

This is called the quotient rule, used when one function is divided by another.

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