In the differential equations slides, the equation f ( x ) + x f ′ ( x ) = h ( x ) is solved using the product rule. Which expression below correctly represents the solution shown in the slides? a) f ( x ) = ∫ h ( x ) d x f(x)=\int h(x)\,dx b) f ( x ) = ∫ h ( x ) d x x f(x)=\dfrac{\int h(x)\,dx}{x} c) f ( x ) = x ∫ h ( x ) d x f(x)=x\int h(x)\,dx d) f ( x ) = h ′ ( x ) + x f(x)=h'(x)+x e) f ( x ) = c e x f(x)=ce^x
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In the calculus slides, it is shown that the rule d d x x n = n x n − 1 also works for fractional, negative, and even zero exponents. However, one function is still “missing” from the table of power derivatives. Which function is it? a) x − 2 x^{-2} b) x − 1 x^{-1} c) ln x \ln x d) e x e^x e) x \sqrt{x}