\(f'(x) = 8 ⋅ 2 ⋅ x^{1} + 1 \text{.}\)

\(f'(x) = 16 x + 1 \text{.}\)

\(f'(p) = -7 ⋅ 6 ⋅ p^{5} + 5 ⋅ 4 ⋅ p^{3} \text{.}\)

\(f'(p) = -42 p^{5} + 20 p^{3} \text{.}\)

\(f'(a) = \frac{3}{7} ⋅ 8 ⋅ a^{7} + 2\frac{1}{2} ⋅ 6 ⋅ a^{5} + \frac{3}{4} ⋅ 4 ⋅ a^{3} + 2\frac{1}{4} ⋅ 3 ⋅ a^{2} \text{.}\)

\(f'(a) = 3\frac{3}{7} a^{7} + 15 a^{5} + 3 a^{3} + 6\frac{3}{4} a^{2} \text{.}\)

(Haakjes wegwerken)
\(f(x) = (8 x^{4} + 2) (x + 6) = 8 x^{5} + 48 x^{4} + 2 x + 12\)

(Differentiëren)
\(f'(x) = 40 x^{4} + 192 x^{3} + 2 \text{.}\)

(Haakjes wegwerken)
\(f(a) = (5 a^{4} + 1)^{2} = 25 a^{8} + 10 a^{4} + 1\)

(Differentiëren)
\(f'(a) = 200 a^{7} + 40 a^{3} \text{.}\)

(Productregel)
\(f'(a) = -4 (9 a^{2} + 4 a) + (-4 a - 3) (18 a + 4) \text{.}\)

(Productregel)
\(f'(x) = (18 x + 6) (7 x^{2} + x + 5) + (9 x^{2} + 6 x) (14 x + 1) \text{.}\)

(Quotiëntregel)
\(f'(x) = {(x + 7) ⋅ -7 - (-7 x - 9) ⋅ 1 \over (x + 7)^{2}} \text{.}\)

\(f'(x) = {(-7 x - 49) - (-7 x - 9) \over (x + 7)^{2}} = {-40 \over (x + 7)^{2}} \text{.}\)

(Quotiëntregel)
\(f'(p) = {(-8 p + 1) ⋅ -6 p - -3 p^{2} ⋅ -8 \over (-8 p + 1)^{2}} \text{.}\)

\(f'(p) = {(48 p^{2} - 6 p) - 24 p^{2} \over (-8 p + 1)^{2}} = {24 p^{2} - 6 p \over (-8 p + 1)^{2}} \text{.}\)

(Herleiden)
\(f(a) = {7 \over 5 a^{2}} = \frac{7}{5} a^{-2}\)

(Differentiëren)
\(f'(a) = \frac{7}{5} ⋅ -2 ⋅ a^{-3} = -\frac{14}{5} ⋅ a^{-3}\)

(Herleiden)
\(f'(a) = -\frac{14}{5} ⋅ {1 \over a^{3}} = -{14 \over 5 a^{3}}\)

(Herleiden)
\(f(a) = -4 a^{3} ⋅ \sqrt[9]{a^{7}} = -4 ⋅ a^{3} ⋅ a^{\frac{7}{9}} = -4 ⋅ a^{3\frac{7}{9}}\)

(Differentiëren)
\(f'(a) = -4 ⋅ 3\frac{7}{9} ⋅ a^{2\frac{7}{9}}\)

(Herleiden)
\(f'(a) = -15\frac{1}{9} ⋅ a^{2} ⋅ a^{\frac{7}{9}} = -15\frac{1}{9} a^{2} ⋅ \sqrt[9]{a^{7}}\)

(Uitdelen)
\(f(p) = {p^{6} \over 5 p^{3}} - {4 p^{2} \over 5 p^{3}} = \frac{1}{5} p^{3} - \frac{4}{5} p^{-1}\)

(Differentiëren)
\(f'(p) = \frac{1}{5} ⋅ 3 ⋅ p^{2} - \frac{4}{5} ⋅ -1 ⋅ p^{-2}\)

(Herleiden)
\(f'(p) = \frac{3}{5} p^{2} + {4 \over 5 p^{2}}\)

(Herleiden)
\(f(x) = {x^{4} + 2 \over x^{\frac{1}{5}}}\)

(Uitdelen)
\(f(x) = {x^{4} \over x^{\frac{1}{5}}} + {2 \over x^{\frac{1}{5}}} = x^{3\frac{4}{5}} + 2 x^{-\frac{1}{5}}\)

(Differentiëren)
\(f'(x) = 3\frac{4}{5} ⋅ x^{2\frac{4}{5}} + 2 ⋅ -\frac{1}{5} ⋅ x^{-1\frac{1}{5}}\)

(Herleiden)
\(f'(x) = 3\frac{4}{5} x^{2} ⋅ \sqrt[5]{x^{4}} - {2 \over 5 x ⋅ \sqrt[5]{x}}\)

(Herleiden)
\(f(x) = {3 \over 8 \sqrt{x}} - 3 \sqrt{x} = \frac{3}{8} x^{-\frac{1}{2}} - 3 x^{\frac{1}{2}}\)

(Differentiëren)
\(f'(x) = \frac{3}{8} ⋅ -\frac{1}{2} ⋅ x^{-1\frac{1}{2}} - 3 ⋅ \frac{1}{2} ⋅ x^{-\frac{1}{2}}\)

(Herleiden)
\(f'(x) = -{3 \over 16 x \sqrt{x}} - {3 \over 2 \sqrt{x}}\)

(Herleiden)
\(f(a) = {2 a - 4 \over a^{3\frac{1}{2}}}\)

(Uitdelen)
\(f(a) = {2 a \over a^{3\frac{1}{2}}} - {4 \over a^{3\frac{1}{2}}} = 2 a^{-2\frac{1}{2}} - 4 a^{-3\frac{1}{2}}\)

(Differentiëren)
\(f'(a) = 2 ⋅ -2\frac{1}{2} ⋅ a^{-3\frac{1}{2}} - 4 ⋅ -3\frac{1}{2} ⋅ a^{-4\frac{1}{2}}\)

(Herleiden)
\(f'(a) = -{5 \over a^{3} ⋅ \sqrt{a}} + {14 \over a^{4} ⋅ \sqrt{a}}\)

(Kettingregel)
\(f'(a) = 8 ⋅ 5 ⋅ (\frac{5}{9} a + 6)^{4} ⋅ \frac{5}{9}\)

(Herleiden)
\(f'(a) = 22\frac{2}{9} (\frac{5}{9} a + 6)^{4} \text{.}\)

(Herleiden)
\(f(a) = {4 \over (3 a - 1)^{5}} = 4 ⋅ (3 a - 1)^{-5}\)

(Kettingregel)
\(f'(a) = 4 ⋅ -5 ⋅ (3 a - 1)^{-6} ⋅ 3\)

(Herleiden)
\(f'(a) = -60 ⋅ (3 a - 1)^{-6} = -{60 \over (3 a - 1)^{6}}\)

(Herleiden)
\(f(p) = 3 \sqrt{5 p + 1} = 3 ⋅ (5 p + 1)^{\frac{1}{2}} \text{.}\)

(Kettingregel)
\(f'(p) = 3 ⋅ \frac{1}{2} ⋅ (5 p + 1)^{-\frac{1}{2}} ⋅ 5\)

(Herleiden)
\(f'(p) = \frac{15}{2} ⋅ (5 p + 1)^{-\frac{1}{2}} = {15 \over 2 \sqrt{5 p + 1}}\)

(Herleiden)
\(f(x) = {9 \over 7 \sqrt{5 x + 3}} = \frac{9}{7} ⋅ (5 x + 3)^{-\frac{1}{2}}\)

(Kettingregel)
\(f'(x) = \frac{9}{7} ⋅ -\frac{1}{2} ⋅ (5 x + 3)^{-1\frac{1}{2}} ⋅ 5\)

(Herleiden)
\(f'(x) = -\frac{45}{14} ⋅ (5 x + 3)^{-1\frac{1}{2}} = -{45 \over 14 (5 x + 3) \sqrt{5 x + 3}}\)

(Kettingregel)
\(f'(x) = 6 ⋅ 4 ⋅ (x^{4} + 2 x^{3} + 3 x^{2})^{3} ⋅ (4 x^{3} + 6 x^{2} + 6 x)\)

(Herleiden)
\(f'(x) = (96 x^{3} + 144 x^{2} + 144 x) ⋅ (x^{4} + 2 x^{3} + 3 x^{2})^{3}\)

\(f(a) = 8 a ⋅ e^{-6 a + 1} + (4 a^{2} - 3) ⋅ e^{-6 a + 1} ⋅ -6\)
\(\text{ } = 8 a ⋅ e^{-6 a + 1} + (-24 a^{2} + 18) ⋅ e^{-6 a + 1}\)
\(\text{ } = (-24 a^{2} + 8 a + 18) ⋅ e^{-6 a + 1}\)

\(f(a) = -2 ⋅ 4^{5 a^{3} - 3 a} ⋅ \ln(4) ⋅ (15 a^{2} - 3) = (-30 a^{2} + 6) ⋅ 4^{5 a^{3} - 3 a} ⋅ \ln(4)\)