Differentiëren

\(f'(p) = 2 ⋅ p^{1} + 3 \text{.}\)

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

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

\(f'(x) = 36 x^{8} + 8 x^{7} - 36 x^{3} \text{.}\)

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

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

(Haakjes wegwerken)
\(f(x) = (5 x^{3} + 9) (x - 2) = 5 x^{4} - 10 x^{3} + 9 x - 18\)

(Differentiëren)
\(f'(x) = 20 x^{3} - 30 x^{2} + 9 \text{.}\)

(Haakjes wegwerken)
\(f(a) = (3 a^{5} - 1)^{2} = 9 a^{10} - 6 a^{5} + 1\)

(Differentiëren)
\(f'(a) = 90 a^{9} - 30 a^{4} \text{.}\)

(Herleiden)
\(f(a) = -6 a^{3} ⋅ \sqrt[9]{a^{2}} = -6 ⋅ a^{3} ⋅ a^{\frac{2}{9}} = -6 ⋅ a^{3\frac{2}{9}}\)

(Differentiëren)
\(f'(a) = -6 ⋅ 3\frac{2}{9} ⋅ a^{2\frac{2}{9}}\)

(Herleiden)
\(f'(a) = -19\frac{1}{3} ⋅ a^{2} ⋅ a^{\frac{2}{9}} = -19\frac{1}{3} a^{2} ⋅ \sqrt[9]{a^{2}}\)

(Herleiden)
\(f(x) = -{7 \over 9 x^{9}} = -\frac{7}{9} x^{-9}\)

(Differentiëren)
\(f'(x) = -\frac{7}{9} ⋅ -9 ⋅ x^{-10} = 7 ⋅ x^{-10}\)

(Herleiden)
\(f'(x) = 7 ⋅ {1 \over x^{10}} = {7 \over x^{10}}\)

(Herleiden)
\(f(a) = {5 \over 3 \sqrt{a}} - 9 \sqrt{a} = \frac{5}{3} a^{-\frac{1}{2}} - 9 a^{\frac{1}{2}}\)

(Differentiëren)
\(f'(a) = \frac{5}{3} ⋅ -\frac{1}{2} ⋅ a^{-1\frac{1}{2}} - 9 ⋅ \frac{1}{2} ⋅ a^{-\frac{1}{2}}\)

(Herleiden)
\(f'(a) = -{5 \over 6 a \sqrt{a}} - {9 \over 2 \sqrt{a}}\)

(Uitdelen)
\(f(x) = {x^{7} \over 4 x^{5}} + {3 x^{2} \over 4 x^{5}} = \frac{1}{4} x^{2} + \frac{3}{4} x^{-3}\)

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

(Herleiden)
\(f'(x) = \frac{1}{2} x - {9 \over 4 x^{4}}\)

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

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

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

(Herleiden)
\(f'(p) = 18\frac{2}{3} p^{3} ⋅ \sqrt[3]{p^{2}} - {1 \over 3 p ⋅ \sqrt[3]{p}}\)

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

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

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

(Herleiden)
\(f'(x) = {5 \over 2 x^{3} ⋅ \sqrt{x}} - {7 \over x^{4} ⋅ \sqrt{x}}\)

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

\(f'(p) = {(-18 p + 6) - (-18 p + 27) \over (-9 p + 3)^{2}} = {-21 \over (-9 p + 3)^{2}} \text{.}\)

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

\(f'(a) = {(16 a^{2} + 10 a) - 8 a^{2} \over (-8 a - 5)^{2}} = {8 a^{2} + 10 a \over (-8 a - 5)^{2}} \text{.}\)

(Kettingregel)
\(f'(x) = 6 ⋅ 3 ⋅ (\frac{2}{7} x - 2)^{2} ⋅ \frac{2}{7}\)

(Herleiden)
\(f'(x) = 5\frac{1}{7} (\frac{2}{7} x - 2)^{2} \text{.}\)

(Kettingregel)
\(f'(p) = 6 ⋅ 3 ⋅ (5 p^{4} + p^{3} + 2 p)^{2} ⋅ (20 p^{3} + 3 p^{2} + 2)\)

(Herleiden)
\(f'(p) = (360 p^{3} + 54 p^{2} + 36) ⋅ (5 p^{4} + p^{3} + 2 p)^{2}\)

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

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

(Herleiden)
\(f'(x) = 60 ⋅ (4 x - 2)^{-4} = {60 \over (4 x - 2)^{4}}\)

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

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

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

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

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

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

\(f(p) = 4 ⋅ e^{2 p^{3} - 3 p^{2}} ⋅ (6 p^{2} - 6 p) = (24 p^{2} - 24 p) ⋅ e^{2 p^{3} - 3 p^{2}}\)

\(f(a) = (6 a^{2} - 2 a) ⋅ e^{-4 a - 2} + (2 a^{3} - a^{2}) ⋅ e^{-4 a - 2} ⋅ -4\)
\(\text{ } = (6 a^{2} - 2 a) ⋅ e^{-4 a - 2} + (-8 a^{3} + 4 a^{2}) ⋅ e^{-4 a - 2}\)
\(\text{ } = (-8 a^{3} + 10 a^{2} - 2 a) ⋅ e^{-4 a - 2}\)

(Productregel)
\(f'(x) = -6 (x^{2} + 2 x) + (-6 x - 3) (2 x + 2) \text{.}\)

(Productregel)
\(f'(a) = (10 a - 9) (-4 a^{2} + 3 a - 5) + (5 a^{2} - 9 a) (-8 a + 3) \text{.}\)