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Chain, product and quotient rules; eˣ, ln x, sin x, cos x
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Learn it Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x). Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v². Also (eˣ)′ = eˣ, (ln x)′ = 1/x, (sin x)′ = cos x, (cos x)′ = −sin x.
Example: Differentiate y = (3x² + 1)⁵.
- Outside: 5(3x² + 1)⁴. Inside: 6x.
- y′ = 30x(3x² + 1)⁴.
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