Answer to Question #227918 in Discrete Mathematics for Val

Question

2.1 Differentiate f(x) = (2x3 + 3x2

)(x2 + 5x3 +5) using product rule. (5)

 2.2 Differentiate f(x) = (8x

3 + 4x )(2x

2 +5) using product rule. (5)

 2.3 Differentiate f(x) = 

5𝑥³+7𝑥

2𝑥²−4𝑥+5

using quotient rule. (5)

 2.4 Differentiate f(x) = 

𝑥³+4

4𝑥− 5

using quotient rule. (5)

 2.5 Differentiate (x5 + 6)5 using chain rule (5)

Answer 1

2.1 Let us differentiate $f(x) = (2x^3 + 3x^2)(x^2 + 5x^3 +5)$ using product rule:

$f'(x) = (6x^2 + 6x)(x^2 + 5x^3 +5)+(2x^3 + 3x^2)(2x + 15x^2)\\ =6x^4+30x^5+30x^2+6x^3+30x^4+30x+4x^4+30x^5+6x^3+45x^4\\ =60x^5+85x^4+12x^3+30x^2+30x$

 2.2 Let us differentiate $f(x) = (8x^3 + 4x )(2x^2 +5)$ using product rule:

$f'(x) = (24x^2 + 4)(2x^2 +5)+(8x^3 + 4x )4x=48x^4+120x^2+8x^2+20+32x^4+16x^2\\ =80x^4+144x^2+20$

 2.3 Let us differentiate $f(x) =\frac{5𝑥^3+7𝑥}{2𝑥^2−4𝑥+5}$ using quotient rule:

$f'(x) =\frac{(15𝑥^2+7)(2𝑥^2−4𝑥+5)-(5𝑥^3+7𝑥)(4x-4)}{(2𝑥^2−4𝑥+5)^2} =\frac{30x^4-60x^3+75x^2+14x^2-28x+35-20x^4+20x^3-28x^2+28x}{(2𝑥^2−4𝑥+5)^2}\\= \frac{10x^4-40x^3+61x^2+35}{(2𝑥^2−4𝑥+5)^2}$

 2.4 Let us differentiate $f(x) =\frac{𝑥^3+4}{4𝑥− 5}$ using quotient rule:

$f'(x) =\frac{3x^2(4x-5)-(𝑥^3+4)4}{(4𝑥− 5)^2} =\frac{8x^3-15x^2-16}{(4𝑥− 5)^2}$

 2.5 Let us differentiate $f(x)=(x^5 + 6)^5$ using chain rule:

$f'(x)=5(x^5 + 6)^4(x^5 + 6)'=25x^4(x^5 + 6)^4.$