Question

Let

Select each of the true statements from the following:

Answerlist

  • For \(h(x)\) the outside function is \(f(x)=\sin(x)\) and the argument is \(g(x)=3x^2\)
  • \(h^'(x) = \cos(6x)\)
  • For \(v(x)\) the outside function is \(f(x)=\sin(x)\) and the argument is \(g(x)=3x^2\)
  • \(v^'(x) = 16.85xe^{16.85x-1}\)
  • For \(r(x)\) the outside function is \(f(x)=8x+11\) and the argument is \(g(x)=\ln(x)\)
  • \(r^'(x) = \frac{8}{8x+11}\)
  • All of the above statements are false.

Solution

Answerlist

  • True. For \(h(x)=\sin(3x^2)\) the outside function is \(f(x)=\sin(x)\) and the argument (or inside function) is \(g(x)=3x^2\). The composition of \(f(x)\) and \(g(x)\) is \[ f(g(x))=f(3x^2)=\sin(3x^2)=h(x) \]
  • False. The chain rule states \[\frac{d}{dx}\sin(3x^2)=h^'(x)=f^'(g(x))g^'(x)\] where the outside function is \(f(x)=\sin(x)\) and the argument (or inside function) is \(g(x)=3x^2\). Since \(f^'(x)=\cos(x)\) (not \(-\cos(x)\)) and \(g^'(x)=6x\) \[h^'(x)=\cos(g(x))\cdot g^'(x)=\cos(3x^2\cdot 6x\]
  • False. The composition of \(f(x)=\sin(x)\) and \(g(x)=3x^2\) is \[ f(g(x))=f(3x^2)=\sin(3x^2) \ne v(x) \]
  • False. What function(s) could have derivative \(16.85xe^{16.85x-1}\)? This question is beyond the scope of this course but the process used to find an answer involves the product rule. Incorrect use of the power rule may be the reason someone would think \[\frac{d}{dx}e^{16.85x} = 16.85xe^{16.85x-1}\]
  • False. The composition of \(f(x)=8x+11\) and \(g(x)=\ln(x)\) is \[ f(g(x))=f(\ln(x))=8\ln(x)+11 \ne r(x) \]
  • True. The chain rule states \[\frac{d}{dx}\ln(8x+11)=r^'(x)=f^'(g(x))g^'(x)\] where the outside function is \(f(x)=\ln(x)\) and the argument (or inside function) is \(g(x)=8x+11\). Since \(f^'(x)=\frac{1}{x}\) and \(g^'(x)=8\) \[r^'(x)=\frac{1}{8x+11}\cdot 8\]
  • False.

Meta-information

extype: mchoice exsolution: 1000010 exname: Chain Rule