Ｃａｌｃｕｌｕｓ　１　ｌｅｃｔｕｒｅ　２．２

Tｅｃｈｎｉｑｕｅｓ　Oｆ　Dｉｆｆｅｒｅｎｔｉａｔｉｏｎ　（ｆｉｎｄｉｎｇ　ｄｅｒｉｖａｔｉｖｅｓ　ｏｆ　ｆｕｎｃｔｉｏｎｓ　ｅａｓｉｌｙ)

I ｎｔｒｏｄｕｃｔｉｏｎ.

• 0:21. Derivative of a Constant. Exploring the slope of a horizontal line – [📷image]
  • Graph of a constant: Horizontal or vertical? 🧩 Example – : $𝒚=c$.
  • The slope of a constant line is zero.
• 1:27. Meaning of "derivative": Represents the slope of a curve.
  • The derivative as the slope of a function.
  • Generalization: the derivative of any constant is zero.
  • The relationship between the slope of a horizontal line and the derivative of a constant is the same $=0$.
• 2:44. In summary: The derivative of a constant is zero.
  • Basic 🧩 Examples –: – [📷image]
    • Derivative of $3$.
    • Derivative of $-1$.
    • Derivative of $\pi$.

Ｆｏｒｍｕｌａ　ｆｏｒ　ｆｉｎｄｉｎｇ　ａ　ｄｅｒｉｖａｔｉｖ.

• 4:10. Using limits to find derivatives when there is a variable.
  • $\dfrac{d}{d𝑥}[𝒇(𝒙)]=\displaystyle \lim_{h\to 0}\dfrac{𝒇(𝒙+h)-𝒇(𝒙)}{h}$
• 4:53. 🧩 Example – : Find the derivative of $𝒇(𝒙)=𝒙^{3}$. – [📷image]
  • Formula for the derivative:
    • $𝒇'(𝒙)=\displaystyle \lim_{h\to 0}\dfrac{𝒇(𝒙+h)-𝒇(𝒙)}{h}$
  • Step 1: Define the functions:
    • $𝒇(𝒙+h)=(𝒙+h)^{3}$
    • $𝒇(𝒙)=𝒙^{3}$
  • Step 2: Substitution into the formula:
    • $𝒇'(𝒙)=\displaystyle \lim_{h\to 0}\dfrac{(𝒙+h)^{3}-𝒙^{3}}{h}$
  • Step 3: Expand the cube:
    • $(𝒙+h)^{3}=𝒙^{3}+3𝒙^{2}h+3𝒙h^{2}+h^{3}$
  • Step 4: Substitution of the expanded expression:
    • $𝒇'(𝒙)=\displaystyle \lim_{h\to 0}\dfrac{(𝒙^{3}+3𝒙^{2}h+3𝒙h^{2}+h^{3})-𝒙^{3}}{h}$
  • Step 5: Simplify the expression:
    • $𝒇'(𝒙)=\displaystyle \lim_{h\to 0}\dfrac{3𝒙^{2}h+3𝒙h^{2}+h^{3}}{h}$
  • Step 6: Factor out $h$:
    • $𝒇'(𝒙)=\displaystyle \lim_{h\to 0}\dfrac{h\big(3𝒙^{2}+3𝒙h+h^{2}\big)}{h}$
  • Step 7: Cancel $h$:
    • $𝒇'(𝒙)=\displaystyle \lim_{h\to 0}\big(3𝒙^{2}+3𝒙h+h^{2}\big)$
  • Step 8: Apply the limit as $h\to 0$:
    • $𝒇'(𝒙)=3𝒙^{2}$
  • Final answer:
    • The derivative of $𝒇(𝒙)=𝒙^{3}$ is $𝒇'(𝒙)=3𝒙^{2}$.
• 7:51. The expression $3𝒙^{2}$ gives the slope of the curve $𝒙^{3}$ at any point.

P ｏｗｅｒ　ｒｕｌｅ　ｆｏｒ　ｄｅｒｉｖａｔｉｖｅｓ

• 8:35. Looking for an easier method to find the derivative.
  • Observing the pattern in the derivative of $𝒙^{3}$.
  • Introduction to the power rule for derivatives: a shorthand method.
• 11:58. ᴩᴏᴡᴇʀ ʀᴜʟᴇ ꜰᴏʀ ᴅᴇʀɪᴠᴀᴛɪᴠᴇꜱ: – [📷image]
  • $\dfrac{d}{d𝑥}[𝒙^{n}]=n\cdot 𝒙^{\,n-1}$, where $n$ is an integer (for now).
• 13:14. 🧩 Examples – of the Power Rule: – [📷image-1] – [📷image-2]
  • $\dfrac{d}{d𝑥}[𝑥^{2}]=2𝑥$.
  • $\dfrac{d}{d𝑥}[𝑥^{5}]=5𝒙^{4}$.
  • $\dfrac{d}{d𝒔}[𝒔^{15}]=15𝒔^{14}$.
  • Adaptation of notation according to the variable.
  • 16:21. Derivatives with Negative Exponents
    • $\dfrac{d}{d𝑥}[𝒙^{-3}]=-3𝒙^{-4}$
    • $\dfrac{d}{d𝒑}[𝒑^{-2}]=-2𝒑^{-3}$
    • $\dfrac{d}{d𝒑}\big[-2𝒑^{-5}\big]=10𝒑^{-6}=\dfrac{10}{𝒑^{6}}$
      • General rule: Rewrite the derivative with positive exponents
    • $\dfrac{d}{d𝑥}\Big[\dfrac{1}{𝒙}\Big]=-\dfrac{1}{𝒙^{2}}$
    • $\dfrac{d}{d𝑥}[𝒙]=1\cdot 𝒙^{0}=1$

D ｅｒｉｖａｔｉｖｅｓ　ｗｉｔｈ　ｃｏｎｓｔａｎｔ　ｍｕｌｔｉｐｌｉｃａｔｉｖｅ　ｆａｃｔｏｒｓ

• 20:25. ʀᴜʟᴇ ꜰᴏʀ ᴅᴇʀɪᴠᴀᴛɪᴠᴇꜱ ᴡɪᴛʜ ᴄᴏɴꜱᴛᴀɴᴛ ᴍᴜʟᴛɪᴩʟɪᴄᴀᴛɪᴠᴇ ꜰᴀᴄᴛᴏʀꜱ: – [📷image]
  • $\dfrac{d}{d𝑥}[𝖼\cdot 𝒇(𝒙)]=𝖼\cdot \dfrac{d}{d𝑥}[𝒇(𝒙)]$, where $𝖼$ is a constant.
• 22:00. 🧩 Example – : $\dfrac{d}{d𝑥}[5𝒙^{4}]=5\cdot \dfrac{d}{d𝑥}[𝒙^{4}]=20𝒙^{3}$.
• 23:13. 🧩 Example – : $\dfrac{d}{d𝑥}[-𝒙^{7}]=-1\cdot \dfrac{d}{d𝑥}[𝒙^{7}]=-7𝑥^{6}$.
• 24:06. 🧩 Example – : $\dfrac{d}{d𝑥}\Big[\dfrac{\pi}{𝒙^{2}}\Big]$.
  • Rewrite the expression to apply the power rule: $\dfrac{d}{d𝑥}\big[\pi\cdot 𝒙^{-2}\big]$.
  • Apply the constant multiplicative rule: $\pi\cdot \dfrac{d}{d𝑥}[𝒙^{-2}]=-2\pi 𝒙^{-3}$.
  • Rewrite with positive exponents: $-\dfrac{2\pi}{𝒙^{3}}$.

D ｅｒｉｖａｔｉｖｅｓ　ｏｆ　ｓｕｍｓ　ａｎｄ　ｄｉｆｆｅｒｅｎｃｅｓ

• 27:25. ʀᴜʟᴇ ꜰᴏʀ ᴅᴇʀɪᴠᴀᴛɪᴠᴇꜱ ᴏꜰ ꜱᴜᴍꜱ ᴀɴᴅ ᴅɪꜰꜰᴇʀᴇɴᴄᴇꜱ: – [📷image]
  • $\dfrac{d}{d𝑥}\big[𝒇(𝒙)\pm 𝓰(𝑥)\big]=\dfrac{d}{d𝑥}[𝒇(𝒙)]\pm \dfrac{d}{d𝑥}[𝓰(𝑥)]$.
• 28:40. 🧩 Example – : $\dfrac{d}{d𝑥}\big[3𝒙^{9}-𝒙^{-3}\big]$.
  • Separate the derivative into two terms: $\dfrac{d}{d𝑥}[3𝑥^{9}]-\dfrac{d}{d𝑥}[𝑥^{-3}]$.
  • Apply the constant multiplicative rule: $3\cdot \dfrac{d}{d𝑥}[𝒙^{9}]-\dfrac{d}{d𝑥}[𝑥^{-3}]$.
  • Result: $27𝒙^{8}-(-3𝒙^{-4})=27𝒙^{8}+\dfrac{3}{𝒙^{4}}$.
• 36:50. 🧩 Example – : $\dfrac{d}{d𝑥}\big[5𝒙^{7}-3𝒙^{4}+2𝒙^{3}+𝒙-1\big]=35𝒙^{6}-12𝒙^{3}+6𝒙^{2}+1$.

D ｅｒｉｖａｔｉｖｅｓ　ｗｉｔｈ　ｆｒａｃｔｉｏｎａｌ　ｅｘｐｏｎｅｎｔｓ

• 32:54. 🧩 Example – : $\dfrac{d}{d𝑥}\big[4-3\sqrt{𝒙}\big]$. – [📷image]
  • Rewrite the root as a fractional exponent: $\dfrac{d}{d𝑥}[4]-\dfrac{d}{d𝑥}\big[3𝒙^{1/2}\big]$.
  • Apply the power rule: $0-\dfrac{3}{2}𝒙^{-1/2}$.

H ｏｒｉｚｏｎｔａｌ　ｔａｎｇｅｎｔ　ｌｉｎｅｓ

• 40:10. Application of derivatives: finding points where a function has horizontal tangent lines. – [📷image]
• 41:00. Horizontal tangent lines have a slope of zero.
• 42:00. To find points with horizontal tangent lines:
  • Find the derivative of the function.
  • Set the derivative equal to zero.
  • Solve for $𝒙$.
• 43:00. 🧩 Example – : Find the points where $𝒚=𝒙^{3}-3𝒙+4$ has horizontal tangent lines.
  • Find the derivative: $\dfrac{d𝒚}{d𝑥}=3𝒙²-3$.
    • 44:25. When a curve’s slope is zero at a point, it signifies a critical point that may be a local or global maximum or minimum. Identifying these points is crucial for optimization and analyzing the behavior of functions in various fiel𝒅𝒔.:
      • Local extrema (maximum or minimum) are the highest or lowest points within a nearby region.
      • Global extrema (maximum or minimum) are the absolute highest or lowest points across the entire domain.
  • Set the derivative to zero: $3𝒙²-3=0$.
  • Solve for $𝒙$: $𝒙=1$; $𝒙=-1$
  • Find the corresponding points on the original function: $(-1,\;6)\text{ and }(1,\;2)$.

H ｉｇｈｅｒ－ｏｒｄｅｒ　ｄｅｒｉｖａｔｉｖｅｓ

• 47:31. Introduction to higher-order derivatives. – [📷image]
• 48:12. Notation for higher-order derivatives:
  • $f''(𝒙)$ for the second derivative.
  • $f'''(𝒙)$ for the third derivative, and so on.
  • $\dfrac{d^{2}𝒚}{d𝑥^{2}}$ for the second derivative of $𝒚$ with respect to $𝑥$.
  • $\dfrac{d^{3}𝒚}{d𝑥^{3}}$ for the third derivative of $𝒚$ with respect to $𝑥$, and so on.
• 52:00. Finding all higher-order derivatives of $7𝒙^{4}-3𝒙^{3}-5𝒙^{2}+9𝒙-347$.
  • First derivative: $28𝒙^{3}-9𝒙^{2}-10𝒙+9$.
  • Second derivative: $84𝒙^{2}-18𝒙-10$.
  • Third derivative: $168𝒙-18$.
  • Fourth derivative: $168$.
  • Fifth derivative: $0$.
  • Sixth derivative and all subsequent derivatives: $0$.
• 56:36. In polynomials, higher-order derivatives eventually become zero.

Ｄｅｒｉｖａｔｉｖｅｓ　ｏｆ　ｃｏｍｐｌｅｘ　ｑｕｏｔｉｅｎｔｓ

• 58:45. 🧩 Example – : $\dfrac{d}{d𝑥}\Big[\dfrac{𝒙^{5}-2𝒙-3}{3\sqrt{𝒙}}\Big]$. – [📷image]
  • Simplify the expression: $\dfrac{𝒙^{5}-2𝒙-3}{3𝒙^{1/2}}$.
  • Cannot take the derivative of each term of the quotient separately.
  • Separate the fraction into three terms: $\dfrac{𝒙^{5}}{3𝒙^{1/2}}-\dfrac{2𝒙}{3𝒙^{1/2}}-\dfrac{3}{3𝒙^{1/2}}$.
  • 1:02:35. Simplify each term using exponent properties.
    • $\dfrac{𝒙^{5}}{3𝒙^{1/2}}=\dfrac{1}{3}𝒙^{9/2}$
    • $\dfrac{2𝒙}{3𝒙^{1/2}}=\dfrac{2}{3}𝒙^{1/2}$
    • $\dfrac{3}{3𝒙^{1/2}}=𝒙^{-1/2}$
  • 1:06:48. Take the derivative of each term separately.
    • Result: $\dfrac{3}{2}𝒙^{7/2}-\dfrac{1}{3}𝒙^{-1/2}+\dfrac{1}{2}𝒙^{-3/2}$
  • 1:10:48. Rewrite the derivative with roots: $\dfrac{3}{2}\sqrt{𝒙^{7}}-\dfrac{1}{3\sqrt{𝒙}}+\dfrac{1}{2\sqrt{𝒙^{3}}}$.