2022 10 04

Section 3.4¶

Problem 3¶

Find the derivative of \(y=e^{9w/8}\) \(\(e^{\frac{9}{8}w}\Rightarrow y'=\tfrac{9}{8}e^{\frac{9w}{8}}\)\)

Problem 6¶

Find the derivative of \(f(w)=(7w^2+8)e^{w^{2}}\) \(\((7w^2+8)e^{w^{2}}\Rightarrow w^{2}(7w^2+8)e^{w^{2}-1}\Rightarrow (7w^4+8w^2)e^{w^{2}-1}\)\) \(f(w)=g(w)*h(w)\) \(f'(w)=g(w)*h'(w)+h(w)*g'(w)\) \(g(w)=(7w^2+8)\) \(h(w)=e^{w^2}\)

\(g'(w)=14w\) \(h'(w)=2we^{w^2}\)

\(f'(w)=(7w^2+8)(2we^{w^2})+(14w)(e^{w^2})\) \(f'(w)=(2we^{w^2})(7w^2+8+7)\) \(f'(w)=(2we^{w^2})(7w^2+15)\)

Problem 10¶

For what values of \(x\) is the graph of \(y=xw^{-3x}\) concave down? (Give your answer as an interval or a list of intervals, e.g.,\((-\infty,8]\textrm{ or }(1,5),(7,10)\))

\(f(x)=xe^{-3x}\) \(f(w)=g(w)*h(w)\) \(f'(w)=g(w)*h'(w)+h(w)*g'(w)\) \(g(x)=x:g'(x)=1\) \(h(x)=e^{-3x}:h'(x)=-3(e^{-3x})\)