Differentiëren

\(f'(p) = 2 ⋅ p^{1} + 5 \text{.}\)

\(f'(p) = 2 p + 5 \text{.}\)

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

\(f'(x) = -48 x^{7} + 21 x^{6} - 25 x^{4} - 8 x \text{.}\)

\(f'(x) = \frac{1}{5} ⋅ 6 ⋅ x^{5} + 1\frac{1}{8} ⋅ 2 ⋅ x^{1} + 1\frac{1}{2} \text{.}\)

\(f'(x) = 1\frac{1}{5} x^{5} + 2\frac{1}{4} x + 1\frac{1}{2} \text{.}\)

(Haakjes wegwerken)
\(f(x) = (3 x^{5} + 9) (x + 6) = 3 x^{6} + 18 x^{5} + 9 x + 54\)

(Differentiëren)
\(f'(x) = 18 x^{5} + 90 x^{4} + 9 \text{.}\)

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

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

(Herleiden)
\(f(p) = -8 p^{2} ⋅ \sqrt[6]{p^{5}} = -8 ⋅ p^{2} ⋅ p^{\frac{5}{6}} = -8 ⋅ p^{2\frac{5}{6}}\)

(Differentiëren)
\(f'(p) = -8 ⋅ 2\frac{5}{6} ⋅ p^{1\frac{5}{6}}\)

(Herleiden)
\(f'(p) = -22\frac{2}{3} ⋅ p^{1} ⋅ p^{\frac{5}{6}} = -22\frac{2}{3} p ⋅ \sqrt[6]{p^{5}}\)

(Herleiden)
\(f(a) = -{3 \over 8 a^{4}} = -\frac{3}{8} a^{-4}\)

(Differentiëren)
\(f'(a) = -\frac{3}{8} ⋅ -4 ⋅ a^{-5} = \frac{3}{2} ⋅ a^{-5}\)

(Herleiden)
\(f'(a) = \frac{3}{2} ⋅ {1 \over a^{5}} = {3 \over 2 a^{5}}\)

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

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

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

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

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

(Herleiden)
\(f'(p) = p - {3 \over p^{3}}\)

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

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

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

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

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

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

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

(Herleiden)
\(f'(x) = -{15 \over 2 x^{2} ⋅ \sqrt{x}} - {10 \over x^{3} ⋅ \sqrt{x}}\)

(Productregel)
\(f'(x) = -1 (-3 x^{2} - 8 x) + (-x - 6) (-6 x - 8) \text{.}\)

(Productregel)
\(f'(a) = (10 a + 4) (-6 a^{2} - a - 5) + (5 a^{2} + 4 a) (-12 a - 1) \text{.}\)

(Quotiëntregel)
\(f'(x) = {(4 x - 7) ⋅ -4 - (-4 x + 3) ⋅ 4 \over (4 x - 7)^{2}} \text{.}\)

\(f'(x) = {(-16 x + 28) - (-16 x + 12) \over (4 x - 7)^{2}} = {16 \over (4 x - 7)^{2}} \text{.}\)

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

\(f'(a) = {(-18 a^{2} + 48 a) - -9 a^{2} \over (3 a - 8)^{2}} = {-9 a^{2} + 48 a \over (3 a - 8)^{2}} \text{.}\)

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

(Herleiden)
\(f'(a) = 18\frac{3}{4} (\frac{5}{8} a + 9)^{5} \text{.}\)

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

(Herleiden)
\(f'(p) = (80 p^{3} + 60) ⋅ (p^{4} + 3 p + 6)^{3}\)

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

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

(Herleiden)
\(f'(x) = 30 ⋅ (3 x + 4)^{-3} = {30 \over (3 x + 4)^{3}}\)

(Herleiden)
\(f(x) = \frac{7}{8} \sqrt{4 x + 3} = \frac{7}{8} ⋅ (4 x + 3)^{\frac{1}{2}} \text{.}\)

(Kettingregel)
\(f'(x) = \frac{7}{8} ⋅ \frac{1}{2} ⋅ (4 x + 3)^{-\frac{1}{2}} ⋅ 4\)

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

(Herleiden)
\(f(a) = {6 \over 5 \sqrt{5 a - 2}} = \frac{6}{5} ⋅ (5 a - 2)^{-\frac{1}{2}}\)

(Kettingregel)
\(f'(a) = \frac{6}{5} ⋅ -\frac{1}{2} ⋅ (5 a - 2)^{-1\frac{1}{2}} ⋅ 5\)

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

\(f(a) = -5 ⋅ e^{-3 a^{3} + 4 a} ⋅ (-9 a^{2} + 4) = (45 a^{2} - 20) ⋅ e^{-3 a^{3} + 4 a}\)

\(f(x) = -12 x^{2} ⋅ e^{5 x - 4} + (-4 x^{3} + 2) ⋅ e^{5 x - 4} ⋅ 5\)
\(\text{ } = -12 x^{2} ⋅ e^{5 x - 4} + (-20 x^{3} + 10) ⋅ e^{5 x - 4}\)
\(\text{ } = (-20 x^{3} - 12 x^{2} + 10) ⋅ e^{5 x - 4}\)