Calculus
# Common Misconceptions (Calculus)

If

\[f(x) = e^{x^2},\]

what is the correct expression for \(f'(x)?\)

Which statement(s) are true?

A. If \(f(x) = x^2,\) then \(f'(x) = 2x.\)

B. If \(f'(x) = 2x,\) then \(f(x) = x^2.\)

True or False?

Suppose \(f\) is a differentiable function on the interval \((a,b).\) If \(f\) is decreasing on \((a,b),\) then \(f'<0\) on \((a,b).\)

True or False?

\[\displaystyle \int_{-1}^1 \frac{1}{x^2} \,dx = \left . \dfrac{x^{-2+1}}{-2+1} \right |_{-1}^1 = -1 -1 = -2. \]

True or False?

If \[f(x) = 2^x,\] then \[f'(x) = x\cdot 2^{x-1}.\]

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