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高二数学 | 高级数学
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13.3 Differentiation Formulas 微分法则


 

[1]

Find the derivative for each of the following functions
求下列各函数的导数

(a) y = 5x^2- 4x + 3

(b) f (x) = 4x^3 + 6x^2 + 2x - 4

(c) y = 4x^3 - 7x^2 + 3x - 5

(d) y = 5x^4 + 3x^3

(e) y = -2x^2 + 4x + 6

(f) f (x) = (2x + 1)(x – 3)

(g) f (x) = (3x - 2)^2

(h) y = (\sqrt{x} - \frac{1}{x})^2

(i) y = \frac{10x^7+2x^2}{x^2}  , x \neq 0

(j) f(x) = \frac{4x+x^3-x^4}{x^2} , x \neq 0

(k) f(x) = \frac{3x^3-4x^2}{\sqrt{x}}

(l) f(x) = \frac{6x^2-2x}{x^4} , x \neq 0

(m) f(x) = \frac{5x + x^2 - \sqrt[3]{x}}{x^3}

(n) f(x) = \sqrt{x} + \frac{1}{\sqrt{x}} , x \neq 0

Answer 答案:
(a) dy/dx=10x – 4

(b) f ‘(x) = 12x^2 + 12x + 2

(c) dy/dx=12x^2 – 14x + 3

(d) dy/dx= 20x^3 + 3x^2

(e) dy/dx= –4x + 4

(f) f ‘(x) = 4x – 5

(g) f ‘ (x) = 18x – 12

(h) dy/dx=1+ \frac{1}{\sqrt{x^3}} - \frac{2}{x^3}

(i) dy/dx= 50x^4

(j) f ‘ (x)=-\frac{4}{x^2} +1-2x

(k) f ‘ (x)=\frac{15}{2} x^{\frac{3}{2}} - 6x^{\frac{1}{2}}

(l) dy/dx= -\frac{12}{x^3} + \frac{6}{x^4}

(m) dy/dx=-10x^{-3} - x^{-2} + \frac{8}{3} x^{-\frac{11}{3}}

(n) dy/dx= \frac{1}{2}x^{-\frac{1}{2}} - \frac{1}{2}x^{-\frac{3}{2}}

[2]

Use the product formula to differentiate the following statements.

用积公式微分下列各式

(a) (2x^3 + 6)(2x^2 + 1)

(b) (2x^3 - 1)(x^2 - 7x + 3)

(c) (5\sqrt{x^3} - 2x)(4x-\frac{3}{x})

(d) ( 2\sqrt{x} + 3 )( \sqrt{x^3} + 2\sqrt{x} - 1 )

Answer 答案:
(a) 20x^4 + 6x^2 + 24x

(b) 10x^4 - 56x^3 + 18x^2 - 2x + 7

(c) 50x^{\frac{3}{2}}- \frac{15}{2} x^{-\frac{1}{2}}-16x

(d) 4x+ \frac{9}{2} \sqrt{x}+ \frac{2}{\sqrt{x}} + 4

[3]

Use the quotient formula to differentiate the following statements.

用商公式微分下列各式

(a) \frac{x-1}{x^2+2}

(b) \frac{2x+1}{2x^3+2x}

(c) \frac{4x^5 + 3(x^2+1)}{3x^2 - 1}

(d) \frac{4x^3- \sqrt[3]{x^2}}{\sqrt{x} + x}

Answer 答案:
(a) \frac{-x^2+2x+2}{(x^2+2)^2}

(b) -\frac{4x^3+3x^2+1}{2x^{2} (x^2+1)^2}

(c) \frac{36x^6-20x^4-24x}{(3x^2-1)^2}

(d) \frac{10x^{\frac{5}{2}}+8x^3- \frac{1}{6} x^{\frac{1}{6}}+\frac{1}{3} x^{\frac{2}{3}}}{(\sqrt{x}+x)^2}