27 exam-style questions on Derivatives of sin, cos, tan, e^x, ln x, x^n (rational); chain/product/quotient; related rates — topic 5.9 of Calculus in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
Determine the derivative of f(x) = cos x.
Answer: −sin x
The derivative of cos x is −sin x; a common slip is dropping the negative sign.
Determine the derivative of f(x) = e^(3x).
Answer: 3e^(3x)
By the chain rule d/dx e^(g(x)) = g'(x)e^(g(x)); here g'(x) = 3, giving 3e^(3x).
Determine the derivative of f(x) = ln(5x).
Answer: 1/x
d/dx ln(kx) = 1/x for any constant k > 0, since the chain rule gives (5)/(5x) = 1/x.
Calculate f'(x) for f(x) = x⁴ using the power rule.
Answer: 4x³
The power rule d/dx xⁿ = n·xⁿ⁻¹ gives 4x³.
A function is given by f(x) = x²·e^x. Determine f'(x) using the product rule.
Answer: e^x(x² + 2x)
Product rule: f'(x) = 2x·e^x + x²·e^x = e^x(x² + 2x).
Determine the derivative of f(x) = sin(2x).
Answer: 2cos(2x)
Chain rule: d/dx sin(2x) = 2·cos(2x).
That was 6 of 27 Derivatives of sin, cos, tan, e^x, ln x, x^n (rational); chain/product/quotient; related rates questions.
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27 Derivatives of sin, cos, tan, e^x, ln x, x^n (rational); chain/product/quotient; related rates questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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