8. Differentiation

Answer: $120 x \left(12 x^{2} + 1\right)^{4}$

Outer power 5, inner $12x^2 + 1$, so $f' = 5(12x^2+1)^{4}\cdot 24x$. This gives $f'(x) = 120 x \left(12 x^{2} + 1\right)^{4}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $5 \left(12 x^{2} + 1\right)^{4}$

Answer: $-2$

$f'(x) = 6 x^{2} + 6 x - 12 = 0$ at $x = -2$ and $x = 1$. $f''(x) = 12 x + 6$, and $f''(-2) = -18 < 0$, so $x = -2$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $1$
• Gave the y-value rather than the x-coordinate → $20$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- x^{2} + 4 x + 7}{x^{4} + 14 x^{2} + 49}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = x - 2$ and $v = x^{2} + 7$: $u' = 1$ and $v' = 2x$, giving $\frac{- x^{2} + 4 x + 7}{x^{4} + 14 x^{2} + 49}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{x^{2} - 4 x - 7}{x^{4} + 14 x^{2} + 49}$
• Differentiated top and bottom separately → $\frac{1}{2 x}$

Answer: $162 x^{5} + 108 x^{3} + 18 x$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $3\left(3 x^{2} + 1\right)^{2} \times 6 x$, which expands to $162 x^{5} + 108 x^{3} + 18 x$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $27 x^{4} + 18 x^{2} + 3$
• Differentiated the inside and raised THAT to the power → $216 x^{3}$

Answer: $x^{5} \left(12 x + 6\right) e^{12 x}$

$u = x^{6}$, $v = e^{12x}$, so $f' = u'v + uv'$. This gives $f'(x) = x^{5} \left(12 x + 6\right) e^{12 x}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $72 x^{5} e^{12 x}$

Answer: $-5$

$f'(x) = 12 x^{2} + 36 x - 120 = 0$ at $x = -5$ and $x = 2$. $f''(x) = 24 x + 36$, and $f''(-5) = -84 < 0$, so $x = -5$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $2$
• Gave the y-value rather than the x-coordinate → $550$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- 3 x^{2} + 4 x + 27}{x^{4} + 18 x^{2} + 81}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = 3 x - 2$ and $v = x^{2} + 9$: $u' = 3$ and $v' = 2x$, giving $\frac{- 3 x^{2} + 4 x + 27}{x^{4} + 18 x^{2} + 81}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{3 x^{2} - 4 x - 27}{x^{4} + 18 x^{2} + 81}$
• Differentiated top and bottom separately → $\frac{3}{2 x}$

Answer: $2048 x^{7} + 12288 x^{5} + 24576 x^{3} + 16384 x$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $4\left(4 x^{2} + 8\right)^{3} \times 8 x$, which expands to $2048 x^{7} + 12288 x^{5} + 24576 x^{3} + 16384 x$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $256 x^{6} + 1536 x^{4} + 3072 x^{2} + 2048$
• Differentiated the inside and raised THAT to the power → $4096 x^{4}$

Answer: $\frac{4 x^{4} \left(x + 10\right)}{x^{2} + 16 x + 64}$

$u = x^{5}$, $v = x + 8$, so $f' = \dfrac{u'v - uv'}{v^2}$. This gives $f'(x) = \frac{4 x^{4} \left(x + 10\right)}{x^{2} + 16 x + 64}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $5 x^{4}$

Answer: $-1$

$f'(x) = 9 x^{2} - 36 x - 45 = 0$ at $x = -1$ and $x = 5$. $f''(x) = 18 x - 36$, and $f''(-1) = -54 < 0$, so $x = -1$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $5$
• Gave the y-value rather than the x-coordinate → $24$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- 4 x^{2} - 12 x + 24}{x^{4} + 12 x^{2} + 36}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = 4 x + 6$ and $v = x^{2} + 6$: $u' = 4$ and $v' = 2x$, giving $\frac{- 4 x^{2} - 12 x + 24}{x^{4} + 12 x^{2} + 36}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{4 x^{2} + 12 x - 24}{x^{4} + 12 x^{2} + 36}$
• Differentiated top and bottom separately → $\frac{2}{x}$

Answer: $8 x^{7} - 192 x^{5} + 1536 x^{3} - 4096 x$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $4\left(x^{2} - 8\right)^{3} \times 2 x$, which expands to $8 x^{7} - 192 x^{5} + 1536 x^{3} - 4096 x$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $4 x^{6} - 96 x^{4} + 768 x^{2} - 2048$
• Differentiated the inside and raised THAT to the power → $16 x^{4}$

Answer: $256 x^{3} + 32 x$

Outer power 2, inner $8x^2 + 1$, so $f' = 2(8x^2+1)^{1}\cdot 16x$. This gives $f'(x) = 256 x^{3} + 32 x$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $16 x^{2} + 2$

Answer: $-3$

$f'(x) = 9 x^{2} - 9 x - 108 = 0$ at $x = -3$ and $x = 4$. $f''(x) = 18 x - 9$, and $f''(-3) = -63 < 0$, so $x = -3$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $4$
• Gave the y-value rather than the x-coordinate → $\frac{405}{2}$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- 6 x^{2} - 6 x + 42}{x^{4} + 14 x^{2} + 49}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = 6 x + 3$ and $v = x^{2} + 7$: $u' = 6$ and $v' = 2x$, giving $\frac{- 6 x^{2} - 6 x + 42}{x^{4} + 14 x^{2} + 49}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{6 x^{2} + 6 x - 42}{x^{4} + 14 x^{2} + 49}$
• Differentiated top and bottom separately → $\frac{3}{x}$

Answer: $750 x^{5} + 1800 x^{3} + 1080 x$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $3\left(5 x^{2} + 6\right)^{2} \times 10 x$, which expands to $750 x^{5} + 1800 x^{3} + 1080 x$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $75 x^{4} + 180 x^{2} + 108$
• Differentiated the inside and raised THAT to the power → $1000 x^{3}$

Answer: $64 x^{3} + 16 x$

Outer power 2, inner $4x^2 + 1$, so $f' = 2(4x^2+1)^{1}\cdot 8x$. This gives $f'(x) = 64 x^{3} + 16 x$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $8 x^{2} + 2$

Answer: $-3$

$f'(x) = 12 x^{2} - 24 x - 180 = 0$ at $x = -3$ and $x = 5$. $f''(x) = 24 x - 24$, and $f''(-3) = -96 < 0$, so $x = -3$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $5$
• Gave the y-value rather than the x-coordinate → $324$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- x^{2} + 18 x + 6}{x^{4} + 12 x^{2} + 36}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = x - 9$ and $v = x^{2} + 6$: $u' = 1$ and $v' = 2x$, giving $\frac{- x^{2} + 18 x + 6}{x^{4} + 12 x^{2} + 36}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{x^{2} - 18 x - 6}{x^{4} + 12 x^{2} + 36}$
• Differentiated top and bottom separately → $\frac{1}{2 x}$

Answer: $320 x^{9} - 2560 x^{7} + 7680 x^{5} - 10240 x^{3} + 5120 x$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $5\left(2 x^{2} - 4\right)^{4} \times 4 x$, which expands to $320 x^{9} - 2560 x^{7} + 7680 x^{5} - 10240 x^{3} + 5120 x$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $80 x^{8} - 640 x^{6} + 1920 x^{4} - 2560 x^{2} + 1280$
• Differentiated the inside and raised THAT to the power → $1024 x^{5}$

Answer: $\frac{x \left(x + 12\right)}{x^{2} + 12 x + 36}$

$u = x^{2}$, $v = x + 6$, so $f' = \dfrac{u'v - uv'}{v^2}$. This gives $f'(x) = \frac{x \left(x + 12\right)}{x^{2} + 12 x + 36}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $2 x$

Answer: $-4$

$f'(x) = 6 x^{2} - 12 x - 144 = 0$ at $x = -4$ and $x = 6$. $f''(x) = 12 x - 12$, and $f''(-4) = -60 < 0$, so $x = -4$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $6$
• Gave the y-value rather than the x-coordinate → $352$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- 3 x^{2} + 12 x + 21}{x^{4} + 14 x^{2} + 49}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = 3 x - 6$ and $v = x^{2} + 7$: $u' = 3$ and $v' = 2x$, giving $\frac{- 3 x^{2} + 12 x + 21}{x^{4} + 14 x^{2} + 49}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{3 x^{2} - 12 x - 21}{x^{4} + 14 x^{2} + 49}$
• Differentiated top and bottom separately → $\frac{3}{2 x}$

Answer: $48 x^{5} - 240 x^{3} + 300 x$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $3\left(2 x^{2} - 5\right)^{2} \times 4 x$, which expands to $48 x^{5} - 240 x^{3} + 300 x$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $12 x^{4} - 60 x^{2} + 75$
• Differentiated the inside and raised THAT to the power → $64 x^{3}$

Answer: $x^{4} \left(10 x + 5\right) e^{10 x}$

$u = x^{5}$, $v = e^{10x}$, so $f' = u'v + uv'$. This gives $f'(x) = x^{4} \left(10 x + 5\right) e^{10 x}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $50 x^{4} e^{10 x}$

Answer: $-7$

$f'(x) = 3 x^{2} + 12 x - 63 = 0$ at $x = -7$ and $x = 3$. $f''(x) = 6 x + 12$, and $f''(-7) = -30 < 0$, so $x = -7$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $3$
• Gave the y-value rather than the x-coordinate → $392$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{4 - 2 x^{2}}{x^{4} + 4 x^{2} + 4}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = 2 x$ and $v = x^{2} + 2$: $u' = 2$ and $v' = 2x$, giving $\frac{4 - 2 x^{2}}{x^{4} + 4 x^{2} + 4}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{2 x^{2} - 4}{x^{4} + 4 x^{2} + 4}$
• Differentiated top and bottom separately → $\frac{1}{x}$

Answer: $8 x^{7}$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $4\left(x^{2}\right)^{3} \times 2 x$, which expands to $8 x^{7}$.

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $4 x^{6}$
• Differentiated the inside and raised THAT to the power → $16 x^{4}$

Answer: $\frac{x \left(x + 18\right)}{x^{2} + 18 x + 81}$

$u = x^{2}$, $v = x + 9$, so $f' = \dfrac{u'v - uv'}{v^2}$. This gives $f'(x) = \frac{x \left(x + 18\right)}{x^{2} + 18 x + 81}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $2 x$

Answer: $-1$

$f'(x) = 12 x^{2} - 12 = 0$ at $x = -1$ and $x = 1$. $f''(x) = 24 x$, and $f''(-1) = -24 < 0$, so $x = -1$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $1$
• Gave the y-value rather than the x-coordinate → $8$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$