Chain Rule Exercise Answers

1. y = e^2x−1

   y = e^u and u = 2x − 1 so = e^u and = 2

   = = e^u2 = 2e^2x−1

2. y = ln (5x −3)

   y = ln u and u = 5x − 3 so = and = 5

   = = 5 = =

3. y = (4 − x)⁷

   y = u⁷ and u = 4 − x so = 7u⁶ and = −1

   = = 7u⁶ (−1) = −7u⁶ = −7(4 − x)⁶

4. y = sin (x + 4)

   y = sin u and u = x + 4 so = cos u and = 1

   = = (cos u)1 = cos u = cos (x + 4)

5. y = e^{cos x}

   y = e^u and u = cos x so = e^u and = −sin x

   = = e^u( −sin x) = (−sin x)e^{cos x}

6. y = ln (5x³ − 12)

   y = ln u and u = 5x³ − 12 so = and = 15x²

   = = 15x² ==

7. y = cos (x⁴)

   y = cos u and u = x⁴ so = −sin u and = 4x³

   = = ( −sin u) 4x³ = −4x³sin (x⁴)

8. y = cos⁴ x

   y = u⁴ and u = cos x so = 4u³ and = −sin x

   = = 4u³ ( −sin x) = (−4sin x)u³ = (−4sin x)cos³x

9. y = ln (3x + 1 )

   y = ln u and u = 3x + 1 so = and = 3

   = = 3 = =

10. y = sin²x

    y = u² and u = sin x so = 2u and = cos x

    = = 2u cos x =2(sin x)(cos x)

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