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 $f'(x) = x\cdot 2^{x-1}.$