Derivatives of exponential and log functions

Answer Key - Grade 12 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

1. 1.

   Ari first writes f(x) = ln((x² + 1)³) as 3 ln(x² + 1), then differentiates. What is f′(2)?

   12/5
2. 2.

   Esme first writes f(x) = ln((x² + 4)⁵) as 5 ln(x² + 4), then differentiates. What is f′(2)?

   5/2
3. 3.

   Jun first writes f(x) = ln((x² + 7)³) as 3 ln(x² + 7), then differentiates. What is f′(3)?

   9/8
4. 4.

   Kira uses the chain rule on h(x) = e^(x²). Which is h′(x)?

   • h′(x) = e^(2x)
   • h′(x) = e^(x²)
   • h′(x) = x²e^(x² − 1)
   • h′(x) = 2xe^(x²)
5. 5.

   Luz uses the chain rule on h(x) = e^(sin x). Which is h′(x)?

   • h′(x) = e^(sin x)
   • h′(x) = sin x · e^(sin x − 1)
   • h′(x) = cos x · e^(sin x)
   • h′(x) = e^(cos x)
6. 6.

   Milo uses the chain rule on h(x) = 2^(3x). Which is h′(x)?

   • h′(x) = 3 · 2^(3x)
   • h′(x) = 3 · 2^(3x) · ln 2
   • h′(x) = 3x · 2^(3x − 1)
   • h′(x) = 2^(3x) · ln 2
7. 7.

   Noah uses the chain rule on h(x) = 5^(x²). Which is h′(x)?

   • h′(x) = 2x · 5^(x²) · ln 5
   • h′(x) = 2x · 5^(x²)
   • h′(x) = 5^(x²) · ln 5
   • h′(x) = x² · 5^(x² − 1)
8. 8.

   Pia writes f(x) = ln(x³√(x + 3)) as 3 ln x + (1/2) ln(x + 3) before differentiating. What is f′(1)?

   25/8
9. 9.

   Tala writes f(x) = ln(x³√(x + 2)) as 3 ln x + (1/2) ln(x + 2) before differentiating. What is f′(2)?

   13/8
10. 10.

    Zee writes f(x) = ln(x³√(x + 8)) as 3 ln x + (1/2) ln(x + 8) before differentiating. What is f′(3)?

    23/22