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# The derivative

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Differential Calculus, limits, slope of tangent line to the curve, the derivative. This slide accompanies my lecture in Differential calculus in LPU Batangas

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### The derivative

1. 1. { The Derivative Slope of a line, secant line, tangent line, slope of a secant line, slope of a line tangent to a curve
2. 2. Lines, slope, point-slope form
3. 3. What do we know about slopes of lines? Lines, slope, point-slope form
4. 4. Lines, slope, point-slope form
5. 5. Lines, slope, point-slope form
6. 6. Lines, slope, point-slope form
7. 7. ο Given a curve, the line that intersects the curve on two points is called a secant line. Secant line
8. 8. ο Given a curve, the line that intersects the curve on two points is called a secant line. Secant line
9. 9. Tangent line
10. 10. ο Given a non-linear curve, how do you get the slope of the curve at a the point (x, f(x) )? Challenge question
11. 11. ο Given a non-linear curve, how do you get the slope of the curve at a the point (x, f(x) )? Challenge question Let π ππΏ be the slope of the tangent line, then the slope of the curve at (x,f(x)) is the same as π ππΏ π ππΏ is the derivative of f at x. πβ² π₯ = π ππΏ π¦ = π(π₯)
12. 12. ο How do we compute π ππΏ? Challenge question
13. 13. How to find π ππΏ π₯π, π π₯π π ππΏ = Ξπ¦ Ξπ₯ = π¦2 β π¦1 π₯2 β π₯2 πβ² π₯ = lim Ξπ₯β0 π π₯ + βπ₯ β π π₯ βπ₯
14. 14. ο Three step rule 1. Find π π₯ + βπ₯ β π π₯ 2. Find π π₯+βπ₯ βπ π₯ βπ₯ 1. Evaluate How to compute for πβ² π₯ lim Ξπ₯β0 π π₯ + βπ₯ β π π₯ βπ₯
15. 15. How to compute for πβ²(π₯) First step
16. 16. How to compute for πβ²(π₯) Second step
17. 17. How to compute for πβ²(π₯) Third step πβ² π₯ = 4π₯ β 3
18. 18. What is πβ² 2 ? πβ² 2 = 4 2 β 3 = 5 What is πβ² 0 ? πβ² 0 = 4 0 β 3 = β3