Parts b and c combined: (These are all the rules I can think of).
Addition rule:
f ( x ) = 1 2 x 6 + 1 2 x 6 f ′ ( x ) = 3 x 5 + 3 x 5 {\displaystyle {\begin{aligned}f(x)&={\frac {1}{2}}x^{6}+{\frac {1}{2}}x^{6}&f'(x)&=3x^{5}+3x^{5}\end{aligned}}}
Subtraction rule:
f ( x ) = 2 x 6 − x 6 f ′ ( x ) = 12 x 5 − 6 x 5 {\displaystyle {\begin{aligned}f(x)&=2x^{6}-x^{6}&f'(x)&=12x^{5}-6x^{5}\end{aligned}}}
Product rule:
f ( x ) = x 2 ⋅ x 4 f ′ ( x ) = 2 x ⋅ x 4 + 4 x 3 ⋅ x 2 {\displaystyle {\begin{aligned}f(x)&=x^{2}\cdot x^{4}&f'(x)&=2x\cdot x^{4}+4x^{3}\cdot x^{2}\end{aligned}}}
Quotient rule:
f ( x ) = x 7 x f ′ ( x ) = 7 x 6 ⋅ x − x 7 ⋅ 1 x 2 {\displaystyle {\begin{aligned}f(x)&={\frac {x^{7}}{x}}&f'(x)&={\frac {7x^{6}\cdot x-x^{7}\cdot 1}{x^{2}}}\end{aligned}}}
Chain rule:
f ( x ) = ( x 3 ) 2 f ′ ( x ) = 3 x 2 ⋅ 2 ( x 3 ) {\displaystyle {\begin{aligned}f(x)&=(x^{3})^{2}&f'(x)&=3x^{2}\cdot 2(x^{3})\end{aligned}}}
Log-diff:
ln ( f ( x ) ) = 6 ln ( x ) f ′ ( x ) = 6 x ⋅ x 6 {\displaystyle {\begin{aligned}\ln(f(x))&=6\ln(x)&f'(x)&={\frac {6}{x}}\cdot x^{6}\end{aligned}}}
Exp-chain rule:
f ( x ) = exp [ 6 ln ( x ) ] f ′ ( x ) = 6 x ⋅ exp [ 6 ln ( x ) ] {\displaystyle {\begin{aligned}f(x)&=\exp[6\ln(x)]&f'(x)&={\frac {6}{x}}\cdot \exp[6\ln(x)]\end{aligned}}}
Limit definition of derivative:
f ′ ( x ) = lim h → 0 ( x + h ) 6 − x 6 h f ′ ( x ) = lim h → 0 6 x 5 h + 15 x 4 h 2 + … + h 6 h f ′ ( x ) = lim a → x a 6 − x 6 a − x f ′ ( x ) = lim a → x ( a − x ) ( a 5 + a 4 x + … x 5 ) ( a − x ) {\displaystyle {\begin{aligned}f'(x)&=\lim _{h\rightarrow 0}{\frac {(x+h)^{6}-x^{6}}{h}}&f'(x)&=\lim _{h\rightarrow 0}{\frac {6x^{5}h+15x^{4}h^{2}+\ldots +h^{6}}{h}}\\f'(x)&=\lim _{a\rightarrow x}{\frac {a^{6}-x^{6}}{a-x}}&f'(x)&=\lim _{a\rightarrow x}{\frac {(a-x)(a^{5}+a^{4}x+\ldots x^{5})}{(a-x)}}\end{aligned}}}
![{\displaystyle {\begin{aligned}f(x)&=\exp[6\ln(x)]&f'(x)&={\frac {6}{x}}\cdot \exp[6\ln(x)]\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/a07ccd892a4454060c3711045bae0457c21356d9)
