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Basic differentiation review

Review the basic differentiation rules and use them to solve problems.

What are the basic differentiation rules?

Sum ruleddx[f(x)+g(x)]=ddxf(x)+ddxg(x)
Difference ruleddx[f(x)g(x)]=ddxf(x)ddxg(x)
Constant multiple ruleddx[kf(x)]=kddxf(x)
Constant ruleddxk=0
The Sum rule says the derivative of a sum of functions is the sum of their derivatives.
The Difference rule says the derivative of a difference of functions is the difference of their derivatives.
The Constant multiple rule says the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
The Constant rule says the derivative of any constant function is always 0.
Want to learn more about basic differentiation rules? Check out this video.

What problems can I solve with basic differentiation rules?

You can find the derivatives of functions that are combinations of other, simpler, functions. For example, H(x) is defined as 2f(x)3g(x)+5. We can find H(x) as follows;
=H(x)=ddxH(x)Equivalent notation=ddx[2f(x)3g(x)+5]Substitute the expression for H(x)=ddx[2f(x)]ddx[3g(x)]+ddx(5)Sum and difference rules=2f(x)3g(x)+0Constant and constant multiple rules
We used the basic differentiation rules to find that H(x)=2f(x)3g(x).
Now suppose we are also given that f(3)=1 and g(3)=5. We can find H(3) as follows:

Check your understanding

Problem 1
x f(x) h(x) f(x) h(x)
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to try more problems like this? Check out this exercise.

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