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Mathematics, five units · Ages 17–18

Differentiating Roots and Exponential Functions Practice

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Designed for 12th grade mathematics students. Rewrite root expressions as fractional powers, differentiate each function using the power or exponential rule, and apply the chain rule wherever an inside function appears.

  1. Find the derivative of f(x) = sqrt(x).

  2. Find the derivative of f(x) = e^x.

  3. Find the derivative of f(x) = 5 * e^(2x).

  4. Find the derivative of f(x) = (3x - 4)^(1/2).

  5. Find the derivative of f(x) = e^(x^2).

  6. Find the derivative of f(x) = sqrt(x^2 + 9).

  7. Find the derivative of f(x) = x * e^x using the product rule.

  8. Find the critical points of f(x) = (x - 2) * e^x by setting f'(x) equal to zero.

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Differentiating Roots and Exponential Functions Practice

Answer key

  1. Find the derivative of f(x) = sqrt(x).

    f'(x) = 1 / (2 * sqrt(x))

  2. Find the derivative of f(x) = e^x.

    f'(x) = e^x

  3. Find the derivative of f(x) = 5 * e^(2x).

    f'(x) = 10 * e^(2x)

  4. Find the derivative of f(x) = (3x - 4)^(1/2).

    f'(x) = 3 / (2 * sqrt(3x - 4))

  5. Find the derivative of f(x) = e^(x^2).

    f'(x) = 2x * e^(x^2)

  6. Find the derivative of f(x) = sqrt(x^2 + 9).

    f'(x) = x / sqrt(x^2 + 9)

  7. Find the derivative of f(x) = x * e^x using the product rule.

    f'(x) = e^x + x * e^x = (x + 1)e^x

  8. Find the critical points of f(x) = (x - 2) * e^x by setting f'(x) equal to zero.

    f'(x) = e^x + (x - 2)e^x = (x - 1)e^x; since e^x is never zero, the only critical point is x = 1.