Solved on Jan 30, 2024

Find derivatives of $s=T x^{2}+7 x G+T^{2}$ with respect to $x$ , $G$ , and $T$ .

STEP 1

Assumptions
1. The function $s$ is given by $s = T x^{2} + 7 x G + T^{2}$ .
2. We need to find the partial derivatives of $s$ with respect to $x$ , $G$ , and $T$ .
3. The variables $x$ , $G$ , and $T$ are independent of each other.

STEP 2

To find the partial derivative of $s$ with respect to $x$ , denoted as $\frac{d s}{d x}$ , we treat all other variables as constants and differentiate $s$ with respect to $x$ .

STEP 3

Differentiate the first term of $s$ with respect to $x$ .
$\frac{d}{d x}(T x^{2}) = 2 T x$

STEP 4

Differentiate the second term of $s$ with respect to $x$ .
$\frac{d}{d x}(7 x G) = 7 G$

STEP 5

Differentiate the third term of $s$ with respect to $x$ .
$\frac{d}{d x}(T^{2}) = 0$

STEP 6

Combine the results from steps 3, 4, and 5 to find the partial derivative of $s$ with respect to $x$ .
$\frac{d s}{d x} = 2 T x + 7 G + 0$

STEP 7

Simplify the expression for $\frac{d s}{d x}$ .
$\frac{d s}{d x} = 2 T x + 7 G$

STEP 8

To find the partial derivative of $s$ with respect to $G$ , denoted as $\frac{d s}{d G}$ , we treat all other variables as constants and differentiate $s$ with respect to $G$ .

STEP 9

Differentiate the first term of $s$ with respect to $G$ .
$\frac{d}{d G}(T x^{2}) = 0$

STEP 10

Differentiate the second term of $s$ with respect to $G$ .
$\frac{d}{d G}(7 x G) = 7 x$

STEP 11

Differentiate the third term of $s$ with respect to $G$ .
$\frac{d}{d G}(T^{2}) = 0$

STEP 12

Combine the results from steps 9, 10, and 11 to find the partial derivative of $s$ with respect to $G$ .
$\frac{d s}{d G} = 0 + 7 x + 0$

STEP 13

Simplify the expression for $\frac{d s}{d G}$ .
$\frac{d s}{d G} = 7 x$

STEP 14

To find the partial derivative of $s$ with respect to $T$ , denoted as $\frac{d s}{d T}$ , we treat all other variables as constants and differentiate $s$ with respect to $T$ .

STEP 15

Differentiate the first term of $s$ with respect to $T$ .
$\frac{d}{d T}(T x^{2}) = x^{2}$

STEP 16

Differentiate the second term of $s$ with respect to $T$ .
$\frac{d}{d T}(7 x G) = 0$

STEP 17

Differentiate the third term of $s$ with respect to $T$ .
$\frac{d}{d T}(T^{2}) = 2 T$

STEP 18

Combine the results from steps 15, 16, and 17 to find the partial derivative of $s$ with respect to $T$ .
$\frac{d s}{d T} = x^{2} + 0 + 2 T$

STEP 19

Simplify the expression for $\frac{d s}{d T}$ .
$\frac{d s}{d T} = x^{2} + 2 T$
The partial derivatives of $s$ are as follows:
(a) $\frac{d s}{d x} = 2 T x + 7 G$
(b) $\frac{d s}{d G} = 7 x$
(c) $\frac{d s}{d T} = x^{2} + 2 T$

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Find derivatives of s=Tx2+7xG+T2s=T x^{2}+7 x G+T^{2}s=Tx2+7xG+T2 with respect to xxx, GGG, and TTT.

STEP 1

Assumptions1. The function sss is given by s=Tx2+7xG+T2s = T x^{2} + 7 x G + T^{2}s=Tx2+7xG+T2.2. We need to find the partial derivatives of sss with respect to xxx, GGG, and TTT.3. The variables xxx, GGG, and TTT are independent of each other.

STEP 2

To find the partial derivative of sss with respect to xxx, denoted as dsdx\frac{d s}{d x}dxds​, we treat all other variables as constants and differentiate sss with respect to xxx.

STEP 3

Differentiate the first term of sss with respect to xxx.ddx(Tx2)=2Tx\frac{d}{d x}(T x^{2}) = 2 T xdxd​(Tx2)=2Tx

STEP 4

Differentiate the second term of sss with respect to xxx.ddx(7xG)=7G\frac{d}{d x}(7 x G) = 7 Gdxd​(7xG)=7G

STEP 5

Differentiate the third term of sss with respect to xxx.ddx(T2)=0\frac{d}{d x}(T^{2}) = 0dxd​(T2)=0

STEP 6

Combine the results from steps 3, 4, and 5 to find the partial derivative of sss with respect to xxx.dsdx=2Tx+7G+0\frac{d s}{d x} = 2 T x + 7 G + 0dxds​=2Tx+7G+0

STEP 7

Simplify the expression for dsdx\frac{d s}{d x}dxds​.dsdx=2Tx+7G\frac{d s}{d x} = 2 T x + 7 Gdxds​=2Tx+7G

STEP 8

To find the partial derivative of sss with respect to GGG, denoted as dsdG\frac{d s}{d G}dGds​, we treat all other variables as constants and differentiate sss with respect to GGG.

STEP 9

Differentiate the first term of sss with respect to GGG.ddG(Tx2)=0\frac{d}{d G}(T x^{2}) = 0dGd​(Tx2)=0

STEP 10

Differentiate the second term of sss with respect to GGG.ddG(7xG)=7x\frac{d}{d G}(7 x G) = 7 xdGd​(7xG)=7x

STEP 11

Differentiate the third term of sss with respect to GGG.ddG(T2)=0\frac{d}{d G}(T^{2}) = 0dGd​(T2)=0

STEP 12

Combine the results from steps 9, 10, and 11 to find the partial derivative of sss with respect to GGG.dsdG=0+7x+0\frac{d s}{d G} = 0 + 7 x + 0dGds​=0+7x+0

STEP 13

Simplify the expression for dsdG\frac{d s}{d G}dGds​.dsdG=7x\frac{d s}{d G} = 7 xdGds​=7x

STEP 14

To find the partial derivative of sss with respect to TTT, denoted as dsdT\frac{d s}{d T}dTds​, we treat all other variables as constants and differentiate sss with respect to TTT.

STEP 15

Differentiate the first term of sss with respect to TTT.ddT(Tx2)=x2\frac{d}{d T}(T x^{2}) = x^{2}dTd​(Tx2)=x2

STEP 16

Differentiate the second term of sss with respect to TTT.ddT(7xG)=0\frac{d}{d T}(7 x G) = 0dTd​(7xG)=0

STEP 17

Differentiate the third term of sss with respect to TTT.ddT(T2)=2T\frac{d}{d T}(T^{2}) = 2 TdTd​(T2)=2T

STEP 18

Combine the results from steps 15, 16, and 17 to find the partial derivative of sss with respect to TTT.dsdT=x2+0+2T\frac{d s}{d T} = x^{2} + 0 + 2 TdTds​=x2+0+2T

STEP 19

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Find derivatives of $s=T x^{2}+7 x G+T^{2}$ with respect to $x$ , $G$ , and $T$ .

Assumptions
1. The function $s$ is given by $s = T x^{2} + 7 x G + T^{2}$ .
2. We need to find the partial derivatives of $s$ with respect to $x$ , $G$ , and $T$ .
3. The variables $x$ , $G$ , and $T$ are independent of each other.

To find the partial derivative of $s$ with respect to $x$ , denoted as $\frac{d s}{d x}$ , we treat all other variables as constants and differentiate $s$ with respect to $x$ .

Differentiate the first term of $s$ with respect to $x$ .
$\frac{d}{d x}(T x^{2}) = 2 T x$

Differentiate the second term of $s$ with respect to $x$ .
$\frac{d}{d x}(7 x G) = 7 G$

Differentiate the third term of $s$ with respect to $x$ .
$\frac{d}{d x}(T^{2}) = 0$

Combine the results from steps 3, 4, and 5 to find the partial derivative of $s$ with respect to $x$ .
$\frac{d s}{d x} = 2 T x + 7 G + 0$

Simplify the expression for $\frac{d s}{d x}$ .
$\frac{d s}{d x} = 2 T x + 7 G$

To find the partial derivative of $s$ with respect to $G$ , denoted as $\frac{d s}{d G}$ , we treat all other variables as constants and differentiate $s$ with respect to $G$ .

Differentiate the first term of $s$ with respect to $G$ .
$\frac{d}{d G}(T x^{2}) = 0$

Differentiate the second term of $s$ with respect to $G$ .
$\frac{d}{d G}(7 x G) = 7 x$

Differentiate the third term of $s$ with respect to $G$ .
$\frac{d}{d G}(T^{2}) = 0$

Combine the results from steps 9, 10, and 11 to find the partial derivative of $s$ with respect to $G$ .
$\frac{d s}{d G} = 0 + 7 x + 0$

Simplify the expression for $\frac{d s}{d G}$ .
$\frac{d s}{d G} = 7 x$

To find the partial derivative of $s$ with respect to $T$ , denoted as $\frac{d s}{d T}$ , we treat all other variables as constants and differentiate $s$ with respect to $T$ .

Differentiate the first term of $s$ with respect to $T$ .
$\frac{d}{d T}(T x^{2}) = x^{2}$

Differentiate the second term of $s$ with respect to $T$ .
$\frac{d}{d T}(7 x G) = 0$

Differentiate the third term of $s$ with respect to $T$ .
$\frac{d}{d T}(T^{2}) = 2 T$

Combine the results from steps 15, 16, and 17 to find the partial derivative of $s$ with respect to $T$ .
$\frac{d s}{d T} = x^{2} + 0 + 2 T$