Differentiëren

\(f'(p) = 3 ⋅ p^{2} + 9 \text{.}\)

\(f'(p) = 3 p^{2} + 9 \text{.}\)

\(f'(p) = -7 ⋅ 8 ⋅ p^{7} - 9 ⋅ 4 ⋅ p^{3} + 7 ⋅ 3 ⋅ p^{2} \text{.}\)

\(f'(p) = -56 p^{7} - 36 p^{3} + 21 p^{2} \text{.}\)

\(f'(a) = 2\frac{1}{3} ⋅ 8 ⋅ a^{7} + \frac{8}{9} ⋅ 6 ⋅ a^{5} + 3 ⋅ 5 ⋅ a^{4} + \frac{1}{5} \text{.}\)

\(f'(a) = 18\frac{2}{3} a^{7} + 5\frac{1}{3} a^{5} + 15 a^{4} + \frac{1}{5} \text{.}\)

(Haakjes wegwerken)
\(f(x) = (6 x^{3} + 7) (x - 4) = 6 x^{4} - 24 x^{3} + 7 x - 28\)

(Differentiëren)
\(f'(x) = 24 x^{3} - 72 x^{2} + 7 \text{.}\)

(Haakjes wegwerken)
\(f(x) = (4 x^{3} + 5)^{2} = 16 x^{6} + 40 x^{3} + 25\)

(Differentiëren)
\(f'(x) = 96 x^{5} + 120 x^{2} \text{.}\)

(Herleiden)
\(f(x) = 3 x ⋅ \sqrt[8]{x^{7}} = 3 ⋅ x^{1} ⋅ x^{\frac{7}{8}} = 3 ⋅ x^{1\frac{7}{8}}\)

(Differentiëren)
\(f'(x) = 3 ⋅ 1\frac{7}{8} ⋅ x^{\frac{7}{8}}\)

(Herleiden)
\(f'(x) = 5\frac{5}{8} ⋅ x^{0} ⋅ x^{\frac{7}{8}} = 5\frac{5}{8} ⋅ \sqrt[8]{x^{7}}\)

(Herleiden)
\(f(x) = -{9 \over 4 x^{2}} = -\frac{9}{4} x^{-2}\)

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

(Herleiden)
\(f'(x) = \frac{9}{2} ⋅ {1 \over x^{3}} = {9 \over 2 x^{3}}\)

(Herleiden)
\(f(p) = {3 \over 7 \sqrt{p}} - 4 \sqrt{p} = \frac{3}{7} p^{-\frac{1}{2}} - 4 p^{\frac{1}{2}}\)

(Differentiëren)
\(f'(p) = \frac{3}{7} ⋅ -\frac{1}{2} ⋅ p^{-1\frac{1}{2}} - 4 ⋅ \frac{1}{2} ⋅ p^{-\frac{1}{2}}\)

(Herleiden)
\(f'(p) = -{3 \over 14 p \sqrt{p}} - {2 \over \sqrt{p}}\)

(Uitdelen)
\(f(a) = {a^{8} \over 3 a^{6}} + {2 a^{3} \over 3 a^{6}} = \frac{1}{3} a^{2} + \frac{2}{3} a^{-3}\)

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

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

(Herleiden)
\(f(p) = {3 p^{4} + 1 \over p^{\frac{1}{5}}}\)

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

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

(Herleiden)
\(f'(p) = 11\frac{2}{5} p^{2} ⋅ \sqrt[5]{p^{4}} - {1 \over 5 p ⋅ \sqrt[5]{p}}\)

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

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

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

(Herleiden)
\(f'(a) = {5 \over 2 a^{3} ⋅ \sqrt{a}} + {14 \over a^{4} ⋅ \sqrt{a}}\)

(Productregel)
\(f'(x) = -9 (2 x^{2} - 7 x) + (-9 x - 3) (4 x - 7) \text{.}\)

(Productregel)
\(f'(x) = (-14 x - 4) (-6 x^{2} - 8 x - 9) + (-7 x^{2} - 4 x) (-12 x - 8) \text{.}\)

(Quotiëntregel)
\(f'(p) = {(p + 6) ⋅ -4 - (-4 p - 1) ⋅ 1 \over (p + 6)^{2}} \text{.}\)

\(f'(p) = {(-4 p - 24) - (-4 p - 1) \over (p + 6)^{2}} = {-23 \over (p + 6)^{2}} \text{.}\)

(Quotiëntregel)
\(f'(p) = {(-8 p + 2) ⋅ -10 p - -5 p^{2} ⋅ -8 \over (-8 p + 2)^{2}} \text{.}\)

\(f'(p) = {(80 p^{2} - 20 p) - 40 p^{2} \over (-8 p + 2)^{2}} = {40 p^{2} - 20 p \over (-8 p + 2)^{2}} \text{.}\)

(Kettingregel)
\(f'(p) = 5 ⋅ 3 ⋅ (\frac{7}{8} p + 1)^{2} ⋅ \frac{7}{8}\)

(Herleiden)
\(f'(p) = 13\frac{1}{8} (\frac{7}{8} p + 1)^{2} \text{.}\)

(Kettingregel)
\(f'(a) = 6 ⋅ 5 ⋅ (4 a^{3} + a + 2)^{4} ⋅ (12 a^{2} + 1)\)

(Herleiden)
\(f'(a) = (360 a^{2} + 30) ⋅ (4 a^{3} + a + 2)^{4}\)

(Herleiden)
\(f(p) = {3 \over (5 p - 2)^{4}} = 3 ⋅ (5 p - 2)^{-4}\)

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

(Herleiden)
\(f'(p) = -60 ⋅ (5 p - 2)^{-5} = -{60 \over (5 p - 2)^{5}}\)

(Herleiden)
\(f(x) = -\frac{3}{4} \sqrt{4 x + 2} = -\frac{3}{4} ⋅ (4 x + 2)^{\frac{1}{2}} \text{.}\)

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

(Herleiden)
\(f'(x) = -\frac{3}{2} ⋅ (4 x + 2)^{-\frac{1}{2}} = -{3 \over 2 \sqrt{4 x + 2}}\)

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

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

(Herleiden)
\(f'(a) = \frac{25}{4} ⋅ (5 a + 2)^{-1\frac{1}{2}} = {25 \over 4 (5 a + 2) \sqrt{5 a + 2}}\)

\(f(x) = -4 ⋅ e^{x + 5} ⋅ 1 = -4 ⋅ e^{x + 5}\)

\(f(a) = 6 a ⋅ e^{-3 a + 2} + (3 a^{2} - 1) ⋅ e^{-3 a + 2} ⋅ -3\)
\(\text{ } = 6 a ⋅ e^{-3 a + 2} + (-9 a^{2} + 3) ⋅ e^{-3 a + 2}\)
\(\text{ } = (-9 a^{2} + 6 a + 3) ⋅ e^{-3 a + 2}\)