QuestionFind the derivatives of these functions:
1. $h(x)=(x^{2}-3x+2)(x^{2}+5x+1)$
2. $h(x)=(5x^{2}+4x+8)(9-x-3x^{2})$
3. $y=\frac{3x^{2}+5x+4}{2x^{2}+6x+3}$
4. $y=\frac{x^{2}+2x+3}{x^{2}-4x+6}$
Show final answers.

Studdy Solution

STEP 1

Assumptions1. We are given four functions to find their derivatives. . We will use the product rule and quotient rule for differentiation.
3. The product rule is given by $(fg)' = f'g + fg'$ and the quotient rule is given by (f/g)' = (gf' - fg') / g^.

STEP 2

Let's start with the first function $h(x)=\left(x^{2}- x+2\right)\left(x^{2}+5 x+1\right)$ . We will use the product rule for differentiation.

STEP 3

First, find the derivative of each part in the function.
$h'(x) = (2x -3)(x^{2}+5 x+1) + (x^{2}-3 x+2)(2x +5)$

STEP 4

implify the above expression to get the derivative of the first function.
$h'(x) =2x^{3} +x^{2} -3x^{2} -15x +2x^{3} +x^{2} -6x^{2} +10x +2x +$

STEP 5

Combine like terms.
$h'(x) =4x^{3} +4x^{2} -4x +5$

STEP 6

Now, let's move to the second function $h(x)=\left(5 x^{2}+4 x+8\right)\left(9-x-3 x^{2}\right)$ . We will use the product rule for differentiation.

STEP 7

First, find the derivative of each part in the function.
$h'(x) = (10x +4)(9 - x -3x^{2}) + (5x^{2} +4x +)(-1 -6x)$

STEP 8

implify the above expression to get the derivative of the second function.
$h'(x) =90x -10x^{2} -30x^{3} +36 -4x -12x^{2} -5x^{2} -4x -8 -30x^{3} -24x -48$

STEP 9

Combine like terms.
$h'(x) = -60x^{3} -29x^{2} +62x -20$

STEP 10

Now, let's move to the third function $y=\frac{3 x^{2}+5 x+4}{2 x^{2}+6 x+3}$ . We will use the quotient rule for differentiation.

STEP 11

First, find the derivative of each part in the function.
$y' = \frac{(x^{} +6x +3)(6x +5) - (3x^{} +5x +4)(4x +6)}{(x^{} +6x +3)^{}}$

STEP 12

implify the above expression to get the derivative of the third function.
$y' = \frac{12x^{} +30x^{2} +18x +10x^{2} +15x +15 -12x^{} -18x^{2} -24x -20x -24}{4x^{4} +24x^{} +36x^{2} +18x +9}$

STEP 13

Combine like terms.
$y' = \frac{20x^{2} +9x -9}{x^{} +24x^{3} +36x^{2} +18x +9}$

STEP 14

Now, let's move to the fourth function $y=\frac{x^{2}+2 x+3}{x^{2}-4 x+6}$ . We will use the quotient rule for differentiation.

STEP 15

First, find the derivative of each part in the function.
$y' = \frac{(x^{2} -4x +)(2x +2) - (x^{2} +2x +3)(2x -4)}{(x^{2} -4x +)^{2}}$

STEP 16

implify the above expression to get the derivative of the fourth function.
$y' = \frac{2x^{3} -8x^{2} +12x +2x^{2} -8x +12 -2x^{3} +8x^{2} -6x -4x^{2} +8x -12}{x^{4} -8x^{3} +24x^{2} -24x +36}$

STEP 17

Combine like terms.
$y' = \frac{-4x^{2} +14x}{x^{4} -x^{3} +24x^{2} -24x +36}$ So the derivatives of the given functions are. $h'(x) =4x^{3} +4x^{2} -4x +5$
2. $h'(x) = -60x^{3} -29x^{2} +62x -20$
3. $y' = \frac{20x^{2} +9x -9}{4x^{4} +24x^{3} +36x^{2} +x +9}$
4. $y' = \frac{-4x^{2} +14x}{x^{4} -x^{3} +24x^{2} -24x +36}$

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QuestionFind the derivatives of these functions:
1. $h(x)=(x^{2}-3x+2)(x^{2}+5x+1)$
2. $h(x)=(5x^{2}+4x+8)(9-x-3x^{2})$
3. $y=\frac{3x^{2}+5x+4}{2x^{2}+6x+3}$
4. $y=\frac{x^{2}+2x+3}{x^{2}-4x+6}$
Show final answers.

Studdy Solution

STEP 1

Assumptions1. We are given four functions to find their derivatives. . We will use the product rule and quotient rule for differentiation.
3. The product rule is given by $(fg)' = f'g + fg'$ and the quotient rule is given by (f/g)' = (gf' - fg') / g^.

STEP 2

Let's start with the first function $h(x)=\left(x^{2}- x+2\right)\left(x^{2}+5 x+1\right)$ . We will use the product rule for differentiation.

STEP 3

First, find the derivative of each part in the function.
$h'(x) = (2x -3)(x^{2}+5 x+1) + (x^{2}-3 x+2)(2x +5)$

STEP 4

implify the above expression to get the derivative of the first function.
$h'(x) =2x^{3} +x^{2} -3x^{2} -15x +2x^{3} +x^{2} -6x^{2} +10x +2x +$

STEP 5

Combine like terms.
$h'(x) =4x^{3} +4x^{2} -4x +5$

STEP 6

Now, let's move to the second function $h(x)=\left(5 x^{2}+4 x+8\right)\left(9-x-3 x^{2}\right)$ . We will use the product rule for differentiation.

STEP 7

First, find the derivative of each part in the function.
$h'(x) = (10x +4)(9 - x -3x^{2}) + (5x^{2} +4x +)(-1 -6x)$

STEP 8

implify the above expression to get the derivative of the second function.
$h'(x) =90x -10x^{2} -30x^{3} +36 -4x -12x^{2} -5x^{2} -4x -8 -30x^{3} -24x -48$

STEP 9

Combine like terms.
$h'(x) = -60x^{3} -29x^{2} +62x -20$

STEP 10

Now, let's move to the third function $y=\frac{3 x^{2}+5 x+4}{2 x^{2}+6 x+3}$ . We will use the quotient rule for differentiation.

STEP 11

First, find the derivative of each part in the function.
$y' = \frac{(x^{} +6x +3)(6x +5) - (3x^{} +5x +4)(4x +6)}{(x^{} +6x +3)^{}}$

STEP 12

implify the above expression to get the derivative of the third function.
$y' = \frac{12x^{} +30x^{2} +18x +10x^{2} +15x +15 -12x^{} -18x^{2} -24x -20x -24}{4x^{4} +24x^{} +36x^{2} +18x +9}$

STEP 13

Combine like terms.
$y' = \frac{20x^{2} +9x -9}{x^{} +24x^{3} +36x^{2} +18x +9}$

STEP 14

Now, let's move to the fourth function $y=\frac{x^{2}+2 x+3}{x^{2}-4 x+6}$ . We will use the quotient rule for differentiation.

STEP 15

First, find the derivative of each part in the function.
$y' = \frac{(x^{2} -4x +)(2x +2) - (x^{2} +2x +3)(2x -4)}{(x^{2} -4x +)^{2}}$

STEP 16

implify the above expression to get the derivative of the fourth function.
$y' = \frac{2x^{3} -8x^{2} +12x +2x^{2} -8x +12 -2x^{3} +8x^{2} -6x -4x^{2} +8x -12}{x^{4} -8x^{3} +24x^{2} -24x +36}$

STEP 17

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Studdy Solution

STEP 1

Assumptions1. We are given four functions to find their derivatives. . We will use the product rule and quotient rule for differentiation.3. The product rule is given by (fg)′=f′g+fg′(fg)' = f'g + fg'(fg)′=f′g+fg′ and the quotient rule is given by (f/g)' = (g*f' - f*g') / g^.

STEP 2

Let's start with the first function h(x)=(x2−x+2)(x2+5x+1)h(x)=\left(x^{2}- x+2\right)\left(x^{2}+5 x+1\right)h(x)=(x2−x+2)(x2+5x+1). We will use the product rule for differentiation.

STEP 3

First, find the derivative of each part in the function.h′(x)=(2x−3)(x2+5x+1)+(x2−3x+2)(2x+5)h'(x) = (2x -3)(x^{2}+5 x+1) + (x^{2}-3 x+2)(2x +5)h′(x)=(2x−3)(x2+5x+1)+(x2−3x+2)(2x+5)

STEP 4

implify the above expression to get the derivative of the first function.h′(x)=2x3+x2−3x2−15x+2x3+x2−6x2+10x+2x+h'(x) =2x^{3} +x^{2} -3x^{2} -15x +2x^{3} +x^{2} -6x^{2} +10x +2x +h′(x)=2x3+x2−3x2−15x+2x3+x2−6x2+10x+2x+

STEP 5

Combine like terms.h′(x)=4x3+4x2−4x+5h'(x) =4x^{3} +4x^{2} -4x +5h′(x)=4x3+4x2−4x+5

STEP 6

Now, let's move to the second function h(x)=(5x2+4x+8)(9−x−3x2)h(x)=\left(5 x^{2}+4 x+8\right)\left(9-x-3 x^{2}\right)h(x)=(5x2+4x+8)(9−x−3x2). We will use the product rule for differentiation.

STEP 7

First, find the derivative of each part in the function.h′(x)=(10x+4)(9−x−3x2)+(5x2+4x+)(−1−6x)h'(x) = (10x +4)(9 - x -3x^{2}) + (5x^{2} +4x +)(-1 -6x)h′(x)=(10x+4)(9−x−3x2)+(5x2+4x+)(−1−6x)

STEP 8

implify the above expression to get the derivative of the second function.h′(x)=90x−10x2−30x3+36−4x−12x2−5x2−4x−8−30x3−24x−48h'(x) =90x -10x^{2} -30x^{3} +36 -4x -12x^{2} -5x^{2} -4x -8 -30x^{3} -24x -48h′(x)=90x−10x2−30x3+36−4x−12x2−5x2−4x−8−30x3−24x−48

STEP 9

Combine like terms.h′(x)=−60x3−29x2+62x−20h'(x) = -60x^{3} -29x^{2} +62x -20h′(x)=−60x3−29x2+62x−20

STEP 10

Now, let's move to the third function y=3x2+5x+42x2+6x+3y=\frac{3 x^{2}+5 x+4}{2 x^{2}+6 x+3}y=2x2+6x+33x2+5x+4​. We will use the quotient rule for differentiation.

STEP 11

First, find the derivative of each part in the function.y′=(x+6x+3)(6x+5)−(3x+5x+4)(4x+6)(x+6x+3)y' = \frac{(x^{} +6x +3)(6x +5) - (3x^{} +5x +4)(4x +6)}{(x^{} +6x +3)^{}}y′=(x+6x+3)(x+6x+3)(6x+5)−(3x+5x+4)(4x+6)​

STEP 12

STEP 13

Combine like terms.y′=20x2+9x−9x+24x3+36x2+18x+9y' = \frac{20x^{2} +9x -9}{x^{} +24x^{3} +36x^{2} +18x +9}y′=x+24x3+36x2+18x+920x2+9x−9​

STEP 14

Now, let's move to the fourth function y=x2+2x+3x2−4x+6y=\frac{x^{2}+2 x+3}{x^{2}-4 x+6}y=x2−4x+6x2+2x+3​. We will use the quotient rule for differentiation.

STEP 15

First, find the derivative of each part in the function.y′=(x2−4x+)(2x+2)−(x2+2x+3)(2x−4)(x2−4x+)2y' = \frac{(x^{2} -4x +)(2x +2) - (x^{2} +2x +3)(2x -4)}{(x^{2} -4x +)^{2}}y′=(x2−4x+)2(x2−4x+)(2x+2)−(x2+2x+3)(2x−4)​

STEP 16

STEP 17

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Assumptions1. We are given four functions to find their derivatives. . We will use the product rule and quotient rule for differentiation.
3. The product rule is given by $(fg)' = f'g + fg'$ and the quotient rule is given by (f/g)' = (gf' - fg') / g^.

Let's start with the first function $h(x)=\left(x^{2}- x+2\right)\left(x^{2}+5 x+1\right)$ . We will use the product rule for differentiation.

First, find the derivative of each part in the function.
$h'(x) = (2x -3)(x^{2}+5 x+1) + (x^{2}-3 x+2)(2x +5)$

implify the above expression to get the derivative of the first function.
$h'(x) =2x^{3} +x^{2} -3x^{2} -15x +2x^{3} +x^{2} -6x^{2} +10x +2x +$

Combine like terms.
$h'(x) =4x^{3} +4x^{2} -4x +5$

Now, let's move to the second function $h(x)=\left(5 x^{2}+4 x+8\right)\left(9-x-3 x^{2}\right)$ . We will use the product rule for differentiation.

First, find the derivative of each part in the function.
$h'(x) = (10x +4)(9 - x -3x^{2}) + (5x^{2} +4x +)(-1 -6x)$

implify the above expression to get the derivative of the second function.
$h'(x) =90x -10x^{2} -30x^{3} +36 -4x -12x^{2} -5x^{2} -4x -8 -30x^{3} -24x -48$

Combine like terms.
$h'(x) = -60x^{3} -29x^{2} +62x -20$

Now, let's move to the third function $y=\frac{3 x^{2}+5 x+4}{2 x^{2}+6 x+3}$ . We will use the quotient rule for differentiation.

First, find the derivative of each part in the function.
$y' = \frac{(x^{} +6x +3)(6x +5) - (3x^{} +5x +4)(4x +6)}{(x^{} +6x +3)^{}}$

Combine like terms.
$y' = \frac{20x^{2} +9x -9}{x^{} +24x^{3} +36x^{2} +18x +9}$

Now, let's move to the fourth function $y=\frac{x^{2}+2 x+3}{x^{2}-4 x+6}$ . We will use the quotient rule for differentiation.

First, find the derivative of each part in the function.
$y' = \frac{(x^{2} -4x +)(2x +2) - (x^{2} +2x +3)(2x -4)}{(x^{2} -4x +)^{2}}$