The derivative of a sum is the sum of the derivatives:
(f + g)' = f' + g'.
When the function has more than two terms, and some weird combination of addition and subtraction, the process is similar. We find the derivatives of the individual terms, combine those derivatives by addition or subtraction as in the original function, and everything works out.
Practice:
Let f(x) = sin x + cos x. Find f'(x).
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We find the derivative of each term of f, then add those derivatives together: The same idea works for differences. The derivative of a difference is the difference of the derivatives: (f - g)' = f ' - g'. | |
Let f(x) = sin x - cos x. Find f ' (x).
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We take the difference of the derivatives of the terms: The fun comes in when we start combining other rules with the addition/subtraction rule. | |
Find the derivative of the polynomial f(x) = 5x^{3 }- 4x^{2}.
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We find the derivative of each piece, then combine. Notice that we're also using the rule for multiplication-by-a-constant! | |
Find the derivative of the polynomial f(x) = 6x^{7} + 5x^{4 }- 3x^{2} + 5. | |
We need to find the derivative of each term, and then combine those derivatives, keeping the addition/subtraction as in the original function. For the sake of organization, find the derivative of each term first: Therefore
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Find the derivative of the function.
Answer
- This function is a line with slope 3, therefore f'(x) = 3. Alternately, we can find the derivatives of the terms and add those together:
Find the derivative of the function.
Answer
- We find the derivatives of the terms and add:
Find the derivative of the function.
Answer
- First we rewrite the function:
Now we can take the derivative:
Find the derivative of the function.
Find the derivative of the function.
Answer
- This is a trick question. Although there's a π and an e and a natural log and a square root, there aren't any x's in the expression
This function is a complicated constant, therefore its derivative is 0!
f'(x) = 0
Find the derivative of the function.
- f(x) = log_{2} x^{3} - log_{2} x^{9}
Answer
- The first thing to do is make the function look better and turn the log_{2} x into :
Now we can take the derivative without too much fuss:
Find the derivative of the function.
- f(x) = log_{2 }x - 2cos x
Answer
- First we rewrite the function:
Then we take the derivative:
Find the derivative of the function.
- f(x) = 3cos x - 2x^{7} + 4x^{3}
Find the derivative of the function.
Answer
- There are no x's in the expression
therefore this is another complicated constant. The derivative of a constant is 0, therefore
f'(x) = 0.
Find the derivative of the function.
- f(x) = x(x + 4) - (x - 1)^{2}
Hint
simplify the function before finding the derivative
Answer
- First we rewrite the function:
All of a sudden the function is simplified f(x) = 6x - 1 describes a straight line with a slope of 6, therefore
f'(x) = 6.