Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified. 1. ƒ(x) = 4 - x2; ƒ′(-3), ƒ′(0), ƒ′(1) 2. F(x) = (x - 1)2 + 1; F′(-1), F′(0), F′(2) 3. g(t) = 1 /t2 ; g′(-1), g′(2), g′(sqrt(3)) 4. k(z) = (1 - z )/2z ; k′(-1), k′(1), k′(sqrt(2)) 5. p(u) = sqrt(3u) ; p′(1), p′(3), p′(2/3) 6. r (s) = sqrt(2s + 1) ; r′(0), r′(1), r′(1/2)
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.
1. ƒ(x) = 4 - x2; ƒ′(-3), ƒ′(0), ƒ′(1)
2. F(x) = (x - 1)2 + 1; F′(-1), F′(0), F′(2)
3. g(t) = 1 /t2 ; g′(-1), g′(2), g′(sqrt(3))
4. k(z) = (1 - z )/2z ; k′(-1), k′(1), k′(sqrt(2))
5. p(u) = sqrt(3u) ; p′(1), p′(3), p′(2/3)
6. r (s) = sqrt(2s + 1) ; r′(0), r′(1), r′(1/2)
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