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.  Given a curve, the line that intersects the curve on two points is called a secant line. Secant line
  8. 8.  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.  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.  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‘š 𝑇𝐿 be the slope of the tangent line, then the slope of the curve at (x,f(x)) is the same as π‘š 𝑇𝐿 π‘š 𝑇𝐿 is the derivative of f at x. 𝑓′ π‘₯ = π‘š 𝑇𝐿 𝑦 = 𝑓(π‘₯)
  12. 12.  How do we compute π‘š 𝑇𝐿? Challenge question
  13. 13. How to find π‘š 𝑇𝐿 π‘₯𝑖, 𝑓 π‘₯𝑖 π‘š 𝑇𝐿 = Δ𝑦 Ξ”π‘₯ = 𝑦2 βˆ’ 𝑦1 π‘₯2 βˆ’ π‘₯2 𝑓′ π‘₯ = lim Ξ”π‘₯β†’0 𝑓 π‘₯ + βˆ†π‘₯ βˆ’ 𝑓 π‘₯ βˆ†π‘₯
  14. 14.  Three step rule 1. Find 𝑓 π‘₯ + βˆ†π‘₯ βˆ’ 𝑓 π‘₯ 2. Find 𝑓 π‘₯+βˆ†π‘₯ βˆ’π‘“ π‘₯ βˆ†π‘₯ 1. Evaluate How to compute for 𝑓′ π‘₯ lim Ξ”π‘₯β†’0 𝑓 π‘₯ + βˆ†π‘₯ βˆ’ 𝑓 π‘₯ βˆ†π‘₯
  15. 15. How to compute for 𝑓′(π‘₯) First step
  16. 16. How to compute for 𝑓′(π‘₯) Second step
  17. 17. How to compute for 𝑓′(π‘₯) Third step 𝑓′ π‘₯ = 4π‘₯ βˆ’ 3
  18. 18. What is 𝑓′ 2 ? 𝑓′ 2 = 4 2 βˆ’ 3 = 5 What is 𝑓′ 0 ? 𝑓′ 0 = 4 0 βˆ’ 3 = βˆ’3

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