Differentiëren

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

\(f'(p) = 8 p + 1 \text{.}\)

\(f'(a) = 7 ⋅ 7 ⋅ a^{6} - 2 ⋅ 4 ⋅ a^{3} + 3 ⋅ a^{2} \text{.}\)

\(f'(a) = 49 a^{6} - 8 a^{3} + 3 a^{2} \text{.}\)

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

\(f'(x) = 1\frac{3}{4} x^{6} + 2 x^{2} \text{.}\)

(Haakjes wegwerken)
\(f(a) = (5 a^{2} + 8) (a + 1) = 5 a^{3} + 5 a^{2} + 8 a + 8\)

(Differentiëren)
\(f'(a) = 15 a^{2} + 10 a + 8 \text{.}\)

(Haakjes wegwerken)
\(f(x) = (4 x^{5} + 3)^{2} = 16 x^{10} + 24 x^{5} + 9\)

(Differentiëren)
\(f'(x) = 160 x^{9} + 120 x^{4} \text{.}\)

(Herleiden)
\(f(x) = -3 x^{2} ⋅ \sqrt[9]{x^{5}} = -3 ⋅ x^{2} ⋅ x^{\frac{5}{9}} = -3 ⋅ x^{2\frac{5}{9}}\)

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

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

(Herleiden)
\(f(x) = -{9 \over 5 x^{9}} = -\frac{9}{5} x^{-9}\)

(Differentiëren)
\(f'(x) = -\frac{9}{5} ⋅ -9 ⋅ x^{-10} = \frac{81}{5} ⋅ x^{-10}\)

(Herleiden)
\(f'(x) = \frac{81}{5} ⋅ {1 \over x^{10}} = {81 \over 5 x^{10}}\)

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

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

(Herleiden)
\(f'(a) = -{1 \over 9 a \sqrt{a}} + {1 \over \sqrt{a}}\)

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

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

(Herleiden)
\(f'(a) = 1\frac{1}{2} a^{2} + {3 \over 2 a^{2}}\)

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

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

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

(Herleiden)
\(f'(p) = 6\frac{2}{3} ⋅ \sqrt[3]{p^{2}} + {1 \over 3 p ⋅ \sqrt[3]{p}}\)

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

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

(Quotiëntregel)
\(f'(p) = {(-6 p + 9) ⋅ 7 - (7 p + 1) ⋅ -6 \over (-6 p + 9)^{2}} \text{.}\)

\(f'(p) = {(-42 p + 63) - (-42 p - 6) \over (-6 p + 9)^{2}} = {69 \over (-6 p + 9)^{2}} \text{.}\)

(Quotiëntregel)
\(f'(x) = {(4 x + 1) ⋅ -6 x - -3 x^{2} ⋅ 4 \over (4 x + 1)^{2}} \text{.}\)

\(f'(x) = {(-24 x^{2} - 6 x) - -12 x^{2} \over (4 x + 1)^{2}} = {-12 x^{2} - 6 x \over (4 x + 1)^{2}} \text{.}\)

(Kettingregel)
\(f'(x) = 9 ⋅ 5 ⋅ (4 x + 1)^{4} ⋅ 4\)

(Herleiden)
\(f'(x) = 180 (4 x + 1)^{4} \text{.}\)

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

(Herleiden)
\(f'(a) = (72 a^{2} + 120) ⋅ (a^{3} + 5 a + 3)^{3}\)

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

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

(Herleiden)
\(f'(a) = -24 ⋅ (3 a + 5)^{-3} = -{24 \over (3 a + 5)^{3}}\)

(Herleiden)
\(f(p) = -2 \sqrt{5 p + 4} = -2 ⋅ (5 p + 4)^{\frac{1}{2}} \text{.}\)

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

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

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

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

(Herleiden)
\(f'(x) = -\frac{21}{8} ⋅ (3 x - 4)^{-1\frac{1}{2}} = -{21 \over 8 (3 x - 4) \sqrt{3 x - 4}}\)

\(f(a) = -5 ⋅ 6^{4 a^{3} + 2 a} ⋅ \ln(6) ⋅ (12 a^{2} + 2) = (-60 a^{2} - 10) ⋅ 6^{4 a^{3} + 2 a} ⋅ \ln(6)\)

\(f(x) = 6 x ⋅ e^{4 x - 6} + (3 x^{2} + 1) ⋅ e^{4 x - 6} ⋅ 4\)
\(\text{ } = 6 x ⋅ e^{4 x - 6} + (12 x^{2} + 4) ⋅ e^{4 x - 6}\)
\(\text{ } = (12 x^{2} + 6 x + 4) ⋅ e^{4 x - 6}\)

(Productregel)
\(f'(x) = -7 (-5 x^{2} + 3 x) + (-7 x + 6) (-10 x + 3) \text{.}\)

(Productregel)
\(f'(a) = (-14 a + 4) (-4 a^{2} + 2 a - 5) + (-7 a^{2} + 4 a) (-8 a + 2) \text{.}\)