QuestionFind the derivatives of these functions using formulas $3, 3, 1-3, 3.6$ :
1. $f(x)=2 x^{2}+3 x-7$
2. $f(x)=x^{5}+2 x^{4}+5 x^{3}+4 x^{2}+x+1$
3. $y=5 x^{2}-4 x+9-\frac{2}{3 x^{3}}+\frac{4}{x^{5}}$
4. $y=(x+7)(5 x-2)$
5. $g(x)=\frac{5 x+3}{7 x+2}$
6. $y=\frac{3-2 x}{4 x-3}$

Studdy Solution

STEP 1

Assumptions1. We are asked to find the derivatives of the given functions. . We will use the power rule for differentiation, which states that the derivative of $x^n$ is $n \cdot x^{n-1}$ .
3. We will use the product rule for differentiation, which states that the derivative of $f(x) \cdot g(x)$ is $f'(x) \cdot g(x) + f(x) \cdot g'(x)$ .
4. We will use the quotient rule for differentiation, which states that the derivative of $\frac{f(x)}{g(x)}$ is \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^}.

STEP 2

Let's start by finding the derivative of the first function $f(x)=2 x^{2}+ x-7$ . We can apply the power rule to each term separately.
$f'(x) = d/dx[2x^2] + d/dx[x] - d/dx[7]$

STEP 3

Calculate the derivative of each term.
$f'(x) =2 \cdot2x^{2-1} +3 \cdot1x^{1-1} -0$

STEP 4

implify the expression.
$f'(x) =4x +3$

STEP 5

Next, find the derivative of the second function $f(x)=x^{5}+2 x^{4}+5 x^{3}+4 x^{2}+x+1$ . Again, we can apply the power rule to each term separately.
$f'(x) = d/dx[x^5] +2d/dx[x^4] +5d/dx[x^3] +4d/dx[x^2] + d/dx[x] + d/dx[1]$

STEP 6

Calculate the derivative of each term.
$f'(x) =5x^{5-1} +2 \cdot4x^{4-1} +5 \cdot3x^{3-1} +4 \cdot2x^{2-1} +1x^{1-1} -0$

STEP 7

implify the expression.
$f'(x) =5x^4 +x^3 +15x^2 +x +1$

STEP 8

Next, find the derivative of the third function $y=5 x^{2}-4 x+-\frac{2}{3 x^{3}}+\frac{4}{x^{5}}$ . We can apply the power rule to each term separately, remembering that the derivative of $1/x^n$ is $-n/x^{n+1}$ .
$y' = d/dx[5x^2] - d/dx[4x] + d/dx[] - d/dx[2/(3x^3)] + d/dx[4/x^5]$

STEP 9

Calculate the derivative of each term.
y' =5 \cdot2x^{2-} -4 \cdotx^{-} - -2 \cdot -3x^{-3-} +4 \cdot -5x^{-5-}

STEP 10

implify the expression.
$y' =10x -4 +2x^{-4} -20x^{-6}$

STEP 11

Next, find the derivative of the fourth function $y=(x+7)(5 x-)$ . Here we need to use the product rule.
$y' = d/dx[(x+7)] \cdot (5x-) + (x+7) \cdot d/dx[(5x-)]$

STEP 12

Calculate the derivative of each term.
$y' = \cdot (5x-2) + (x+7) \cdot5$

STEP 13

implify the expression.
$y' =5x -2 +5x +35$

STEP 14

Combine like terms.
$y' =10x +33$

STEP 15

Next, find the derivative of the fifth function $g(x)=\frac{5 x+3}{7 x+2}$ . Here we need to use the quotient rule.
$g'(x) = \frac{d/dx[(5x+3)] \cdot (7x+2) - (5x+3) \cdot d/dx[(7x+2)]}{(7x+2)^2}$

STEP 16

Calculate the derivative of each term.
$g'(x) = \frac{5 \cdot (x+2) - (5x+3) \cdot}{(x+2)^2}$

STEP 17

implify the expression.
$g'(x) = \frac{35x +10 -35x -21}{(7x+2)^2}$

STEP 18

Combine like terms.
$g'(x) = \frac{-11}{(7x+2)^2}$

STEP 19

Finally, find the derivative of the sixth function $y=\frac{3- x}{4 x-3}$ . Again, we need to use the quotient rule.
y' = \frac{d/dx[(3-x)] \cdot (4x-3) - (3-x) \cdot d/dx[(4x-3)]}{(4x-3)^}

STEP 20

Calculate the derivative of each term.
y' = \frac{- \cdot (4x-3) - (3-x) \cdot4}{(4x-3)^}

STEP 21

implify the expression.
y' = \frac{-8x +6 -12 +8x}{(4x-3)^}

STEP 22

Combine like terms.
y' = \frac{-6}{(4x-)^}The derivatives of the given functions are1. $f'(x) =4x +$ . $f'(x) =5x^4 +8x^ +15x^ +8x +1$ . $y' =10x -4 +x^{-4} -20x^{-6}$
4. $y' =10x +33$
5. g'(x) = \frac{-11}{(7x+)^}
6. y' = \frac{-6}{(4x-)^}

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Studdy Solution

STEP 1

STEP 2

Let's start by finding the derivative of the first function f(x)=2x2+x−7f(x)=2 x^{2}+ x-7f(x)=2x2+x−7. We can apply the power rule to each term separately.f′(x)=d/dx[2x2]+d/dx[x]−d/dx[7]f'(x) = d/dx[2x^2] + d/dx[x] - d/dx[7]f′(x)=d/dx[2x2]+d/dx[x]−d/dx[7]

STEP 3

Calculate the derivative of each term.f′(x)=2⋅2x2−1+3⋅1x1−1−0f'(x) =2 \cdot2x^{2-1} +3 \cdot1x^{1-1} -0f′(x)=2⋅2x2−1+3⋅1x1−1−0

STEP 4

implify the expression.f′(x)=4x+3f'(x) =4x +3f′(x)=4x+3

STEP 5

STEP 6

Calculate the derivative of each term.f′(x)=5x5−1+2⋅4x4−1+5⋅3x3−1+4⋅2x2−1+1x1−1−0f'(x) =5x^{5-1} +2 \cdot4x^{4-1} +5 \cdot3x^{3-1} +4 \cdot2x^{2-1} +1x^{1-1} -0f′(x)=5x5−1+2⋅4x4−1+5⋅3x3−1+4⋅2x2−1+1x1−1−0

STEP 7

implify the expression.f′(x)=5x4+x3+15x2+x+1f'(x) =5x^4 +x^3 +15x^2 +x +1f′(x)=5x4+x3+15x2+x+1

STEP 8

STEP 9

Calculate the derivative of each term.y' =5 \cdot2x^{2-} -4 \cdotx^{-} - -2 \cdot -3x^{-3-} +4 \cdot -5x^{-5-}

STEP 10

implify the expression.y′=10x−4+2x−4−20x−6y' =10x -4 +2x^{-4} -20x^{-6}y′=10x−4+2x−4−20x−6

STEP 11

Next, find the derivative of the fourth function y=(x+7)(5x−)y=(x+7)(5 x-)y=(x+7)(5x−). Here we need to use the product rule.y′=d/dx[(x+7)]⋅(5x−)+(x+7)⋅d/dx[(5x−)]y' = d/dx[(x+7)] \cdot (5x-) + (x+7) \cdot d/dx[(5x-)]y′=d/dx[(x+7)]⋅(5x−)+(x+7)⋅d/dx[(5x−)]

STEP 12

Calculate the derivative of each term.y′=⋅(5x−2)+(x+7)⋅5y' = \cdot (5x-2) + (x+7) \cdot5y′=⋅(5x−2)+(x+7)⋅5

STEP 13

implify the expression.y′=5x−2+5x+35y' =5x -2 +5x +35y′=5x−2+5x+35

STEP 14

Combine like terms.y′=10x+33y' =10x +33y′=10x+33

STEP 15

STEP 16

Calculate the derivative of each term.g′(x)=5⋅(x+2)−(5x+3)⋅(x+2)2g'(x) = \frac{5 \cdot (x+2) - (5x+3) \cdot}{(x+2)^2}g′(x)=(x+2)25⋅(x+2)−(5x+3)⋅​

STEP 17

implify the expression.g′(x)=35x+10−35x−21(7x+2)2g'(x) = \frac{35x +10 -35x -21}{(7x+2)^2}g′(x)=(7x+2)235x+10−35x−21​

STEP 18

Combine like terms.g′(x)=−11(7x+2)2g'(x) = \frac{-11}{(7x+2)^2}g′(x)=(7x+2)2−11​

STEP 19

Finally, find the derivative of the sixth function y=3−x4x−3y=\frac{3- x}{4 x-3}y=4x−33−x​. Again, we need to use the quotient rule.y' = \frac{d/dx[(3-x)] \cdot (4x-3) - (3-x) \cdot d/dx[(4x-3)]}{(4x-3)^}

STEP 20

Calculate the derivative of each term.y' = \frac{- \cdot (4x-3) - (3-x) \cdot4}{(4x-3)^}

STEP 21

implify the expression.y' = \frac{-8x +6 -12 +8x}{(4x-3)^}

STEP 22

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Let's start by finding the derivative of the first function $f(x)=2 x^{2}+ x-7$ . We can apply the power rule to each term separately.
$f'(x) = d/dx[2x^2] + d/dx[x] - d/dx[7]$

Calculate the derivative of each term.
$f'(x) =2 \cdot2x^{2-1} +3 \cdot1x^{1-1} -0$

implify the expression.
$f'(x) =4x +3$

Calculate the derivative of each term.
$f'(x) =5x^{5-1} +2 \cdot4x^{4-1} +5 \cdot3x^{3-1} +4 \cdot2x^{2-1} +1x^{1-1} -0$

implify the expression.
$f'(x) =5x^4 +x^3 +15x^2 +x +1$

Calculate the derivative of each term.
y' =5 \cdot2x^{2-} -4 \cdotx^{-} - -2 \cdot -3x^{-3-} +4 \cdot -5x^{-5-}

implify the expression.
$y' =10x -4 +2x^{-4} -20x^{-6}$

Next, find the derivative of the fourth function $y=(x+7)(5 x-)$ . Here we need to use the product rule.
$y' = d/dx[(x+7)] \cdot (5x-) + (x+7) \cdot d/dx[(5x-)]$

Calculate the derivative of each term.
$y' = \cdot (5x-2) + (x+7) \cdot5$

implify the expression.
$y' =5x -2 +5x +35$

Combine like terms.
$y' =10x +33$

Calculate the derivative of each term.
$g'(x) = \frac{5 \cdot (x+2) - (5x+3) \cdot}{(x+2)^2}$

implify the expression.
$g'(x) = \frac{35x +10 -35x -21}{(7x+2)^2}$

Combine like terms.
$g'(x) = \frac{-11}{(7x+2)^2}$

Finally, find the derivative of the sixth function $y=\frac{3- x}{4 x-3}$ . Again, we need to use the quotient rule.
y' = \frac{d/dx[(3-x)] \cdot (4x-3) - (3-x) \cdot d/dx[(4x-3)]}{(4x-3)^}

Calculate the derivative of each term.
y' = \frac{- \cdot (4x-3) - (3-x) \cdot4}{(4x-3)^}

implify the expression.
y' = \frac{-8x +6 -12 +8x}{(4x-3)^}