Differentiate (sin θ)ᶜᵒˢ ᶿ and (𝑥 + sec 𝑥)⁴ᐟˣ

𝗤𝗨𝗘𝗦𝗧𝗜𝗢𝗡 Differentiate
0:03 𝑟(θ) = (sin θ)ᶜᵒˢ ᶿ
2:40 𝑦(𝑥) = (𝑥 + sec 𝑥)⁴ᐟˣ
𝗪𝗢𝗥𝗞𝗙𝗟𝗢𝗪 When a variable appears in both the base as well as the exponent, use 𝗹𝗼𝗴𝗮𝗿𝗶𝘁𝗵𝗺𝗶𝗰 𝗱𝗶𝗳𝗳𝗲𝗿𝗲𝗻𝘁𝗶𝗮𝘁𝗶𝗼𝗻. Take the natural log of the entire equation to bring down the exponent. Differentiate by the Product Rule or Quotient Rule. A summary of the steps is as follows:
① Take the natural logarithm (ln) of both sides of the equation.
② Bring the exponent down, and make the equation look like a product or a quotient.
③ Differentiate by the Product Rule or Quotient Rule. Remember that the derivative of ln is a reciprocal (1/something). Then, by the Chain Rule, multiply by the derivative of any inside function(s). When differentiating the dependent variable (here, 𝑟 or 𝑦), implicit differentiation says we have to multiply by 𝚍𝑟/𝚍θ or 𝚍𝑦/𝚍𝑥.
④ Algebraically, isolate 𝚍𝑟/𝚍θ or 𝚍𝑦/𝚍𝑥 by multiplying both sides of the derivative by 𝑟 or 𝑦.
⑤ Rewrite 𝑟 or 𝑦 in terms of θ or 𝑥, according to the question’s original equation 𝑟(θ) or 𝑦(𝑥).
💡 Memorize the property of logarithms to apply in logarithmic differentiation: log(𝑎)ⁿ = 𝑛 log(𝑎)
⚠️ 𝗔𝗧𝗧𝗘𝗡𝗧𝗜𝗢𝗡
0:55 Cosine differentiates to −sine. Remember to put on the negative sign.
1:35 The entire right side is multiplied by 𝑟. Use brackets to enclose both terms on the right side.
Original content © 2024 Jung-Lynn Jonathan Yang, CC-BY-NC-ND
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