Science:Math Exam Resources/Courses/MATH110/December 2013/Question 03 (b)/Solution 1

Parts b and c combined: (These are all the rules I can think of).

• Addition rule:

  $\begin{array}{rlrl}f(x)& =\frac{1}{2}{x}^6+\frac{1}{2}{x}^6& f^{\prime }(x)& =3x^5+3x^5\end{array}$

• Subtraction rule:

  $\begin{array}{rlrl}f(x)& =2x^6-x^6& f^{\prime }(x)& =12x^5-6x^5\end{array}$

• Product rule:

  $\begin{array}{rlrl}f(x)& =x^2\cdot x^4& f^{\prime }(x)& =2x\cdot x^4+4x^3\cdot x^2\end{array}$

• Quotient rule:

  $\begin{array}{rlrl}f(x)& =\frac{x^7}{x}& f^{\prime }(x)& =\frac{7x^6\cdot x-x^7\cdot 1}{x^2}\end{array}$

• Chain rule:

  $\begin{array}{rlrl}f(x)& =(x^3{)}^2& f^{\prime }(x)& =3x^2\cdot 2(x^3)\end{array}$

• Log-diff:

  $\begin{array}{rlrl}\ln (f(x))& =6\ln (x)& f^{\prime }(x)& =\frac{6}{x}\cdot x^6\end{array}$

• Exp-chain rule:

  $\begin{array}{rlrl}f(x)& =\exp [6\ln (x)]& f^{\prime }(x)& =\frac{6}{x}\cdot \exp [6\ln (x)]\end{array}$

• Limit definition of derivative:

  $\begin{array}{rlrl}{f}^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{(x+h{)}^6-x^6}{h}& f^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{6x^{5}h+15x^4{h}^2+\dots +h^6}{h}\\ f^{\prime }(x)& =\underset{a\to x}{\lim }\frac{a^6-x^6}{a-x}& f^{\prime }(x)& =\underset{a\to x}{\lim }\frac{(a-x)(a^5+a^{4}x+\dots x^5)}{(a-x)}\end{array}$