Calculus Examples

$x = 6 {(3 y - 8)}^{4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x) = \frac{d}{d x} (6 {(3 y - 8)}^{4})$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the Binomial Theorem.

$\frac{d}{d x} [6 ({(3 y)}^{4} + 4 {(3 y)}^{3} \cdot - 8 + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2

Differentiate.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the product rule to $3 y$ .

$\frac{d}{d x} [6 (3^{4} y^{4} + 4 {(3 y)}^{3} \cdot - 8 + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.2

Raise $3$ to the power of $4$ .

$\frac{d}{d x} [6 (81 y^{4} + 4 {(3 y)}^{3} \cdot - 8 + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.3

Apply the product rule to $3 y$ .

$\frac{d}{d x} [6 (81 y^{4} + 4 (3^{3} y^{3}) \cdot - 8 + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.4

Raise $3$ to the power of $3$ .

$\frac{d}{d x} [6 (81 y^{4} + 4 (27 y^{3}) \cdot - 8 + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.5

Multiply $27$ by $4$ .

$\frac{d}{d x} [6 (81 y^{4} + 108 y^{3} \cdot - 8 + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.6

Multiply $- 8$ by $108$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 6 {(3 y)}^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.7

Apply the product rule to $3 y$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 6 (3^{2} y^{2}) {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.8

Raise $3$ to the power of $2$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 6 (9 y^{2}) {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.9

Multiply $9$ by $6$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 54 y^{2} {(- 8)}^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.10

Raise $- 8$ to the power of $2$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 54 y^{2} \cdot 64 + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.11

Multiply $64$ by $54$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 3456 y^{2} + 4 (3 y) {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.12

Multiply $3$ by $4$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 3456 y^{2} + 12 y {(- 8)}^{3} + {(- 8)}^{4})]$

Step 3.2.1.13

Raise $- 8$ to the power of $3$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 3456 y^{2} + 12 y \cdot - 512 + {(- 8)}^{4})]$

Step 3.2.1.14

Multiply $- 512$ by $12$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 3456 y^{2} - 6144 y + {(- 8)}^{4})]$

Step 3.2.1.15

Raise $- 8$ to the power of $4$ .

$\frac{d}{d x} [6 (81 y^{4} - 864 y^{3} + 3456 y^{2} - 6144 y + 4096)]$

Step 3.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 (81 y^{4} - 864 y^{3} + 3456 y^{2} - 6144 y + 4096)$ with respect to $x$ is $6 \frac{d}{d x} [81 y^{4} - 864 y^{3} + 3456 y^{2} - 6144 y + 4096]$ .

$6 \frac{d}{d x} [81 y^{4} - 864 y^{3} + 3456 y^{2} - 6144 y + 4096]$

Step 3.2.3

By the Sum Rule, the derivative of $81 y^{4} - 864 y^{3} + 3456 y^{2} - 6144 y + 4096$ with respect to $x$ is $\frac{d}{d x} [81 y^{4}] + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096]$ .

$6 (\frac{d}{d x} [81 y^{4}] + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.2.4

Since $81$ is constant with respect to $x$ , the derivative of $81 y^{4}$ with respect to $x$ is $81 \frac{d}{d x} [y^{4}]$ .

$6 (81 \frac{d}{d x} [y^{4}] + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = y$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$6 (81 (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [y]) + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$6 (81 (4 {u_{1}}^{3} \frac{d}{d x} [y]) + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $y$ .

$6 (81 (4 y^{3} \frac{d}{d x} [y]) + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.4

Multiply $4$ by $81$ .

$6 (324 (y^{3} \frac{d}{d x} [y]) + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.5

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 (324 y^{3} y' + \frac{d}{d x} [- 864 y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.6

Since $- 864$ is constant with respect to $x$ , the derivative of $- 864 y^{3}$ with respect to $x$ is $- 864 \frac{d}{d x} [y^{3}]$ .

$6 (324 y^{3} y' - 864 \frac{d}{d x} [y^{3}] + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 3.7.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$6 (324 y^{3} y' - 864 (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [y]) + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$6 (324 y^{3} y' - 864 (3 {u_{2}}^{2} \frac{d}{d x} [y]) + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.7.3

Replace all occurrences of $u_{2}$ with $y$ .

$6 (324 y^{3} y' - 864 (3 y^{2} \frac{d}{d x} [y]) + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.8

Multiply $3$ by $- 864$ .

$6 (324 y^{3} y' - 2592 (y^{2} \frac{d}{d x} [y]) + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.9

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + \frac{d}{d x} [3456 y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.10

Since $3456$ is constant with respect to $x$ , the derivative of $3456 y^{2}$ with respect to $x$ is $3456 \frac{d}{d x} [y^{2}]$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 3456 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.11

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.11.1

To apply the Chain Rule, set $u_{3}$ as $y$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 3456 (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.11.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 3456 (2 u_{3} \frac{d}{d x} [y]) + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.11.3

Replace all occurrences of $u_{3}$ with $y$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 3456 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.12

Multiply $2$ by $3456$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 6912 (y \frac{d}{d x} [y]) + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.13

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 6912 y y' + \frac{d}{d x} [- 6144 y] + \frac{d}{d x} [4096])$

Step 3.14

Since $- 6144$ is constant with respect to $x$ , the derivative of $- 6144 y$ with respect to $x$ is $- 6144 \frac{d}{d x} [y]$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 6912 y y' - 6144 \frac{d}{d x} [y] + \frac{d}{d x} [4096])$

Step 3.15

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 6912 y y' - 6144 y' + \frac{d}{d x} [4096])$

Step 3.16

Since $4096$ is constant with respect to $x$ , the derivative of $4096$ with respect to $x$ is $0$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 6912 y y' - 6144 y' + 0)$

Step 3.17

Add $324 y^{3} y' - 2592 y^{2} y' + 6912 y y' - 6144 y'$ and $0$ .

$6 (324 y^{3} y' - 2592 y^{2} y' + 6912 y y' - 6144 y')$

Step 3.18

Simplify.

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Step 3.18.1

Apply the distributive property.

$6 (324 y^{3} y') + 6 (- 2592 y^{2} y') + 6 (6912 y y') + 6 (- 6144 y')$

Step 3.18.2

Combine terms.

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Step 3.18.2.1

Multiply $324$ by $6$ .

$1944 (y^{3} y') + 6 (- 2592 y^{2} y') + 6 (6912 y y') + 6 (- 6144 y')$

Step 3.18.2.2

Multiply $- 2592$ by $6$ .

$1944 y^{3} y' - 15552 (y^{2} y') + 6 (6912 y y') + 6 (- 6144 y')$

Step 3.18.2.3

Multiply $6912$ by $6$ .

$1944 y^{3} y' - 15552 y^{2} y' + 41472 (y y') + 6 (- 6144 y')$

Step 3.18.2.4

Multiply $- 6144$ by $6$ .

$1944 y^{3} y' - 15552 y^{2} y' + 41472 y y' - 36864 y'$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = 1944 y^{3} y' - 15552 y^{2} y' + 41472 y y' - 36864 y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $1944 y^{3} y' - 15552 y^{2} y' + 41472 y y' - 36864 y' = 1$ .

$1944 y^{3} y' - 15552 y^{2} y' + 41472 y y' - 36864 y' = 1$

Step 5.2

Factor $72 y'$ out of $1944 y^{3} y' - 15552 y^{2} y' + 41472 y y' - 36864 y'$ .

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Step 5.2.1

Factor $72 y'$ out of $1944 y^{3} y'$ .

$72 y' (27 y^{3}) - 15552 y^{2} y' + 41472 y y' - 36864 y' = 1$

Step 5.2.2

Factor $72 y'$ out of $- 15552 y^{2} y'$ .

$72 y' (27 y^{3}) + 72 y' (- 216 y^{2}) + 41472 y y' - 36864 y' = 1$

Step 5.2.3

Factor $72 y'$ out of $41472 y y'$ .

$72 y' (27 y^{3}) + 72 y' (- 216 y^{2}) + 72 y' (576 y) - 36864 y' = 1$

Step 5.2.4

Factor $72 y'$ out of $- 36864 y'$ .

$72 y' (27 y^{3}) + 72 y' (- 216 y^{2}) + 72 y' (576 y) + 72 y' \cdot - 512 = 1$

Step 5.2.5

Factor $72 y'$ out of $72 y' (27 y^{3}) + 72 y' (- 216 y^{2})$ .

$72 y' (27 y^{3} - 216 y^{2}) + 72 y' (576 y) + 72 y' \cdot - 512 = 1$

Step 5.2.6

Factor $72 y'$ out of $72 y' (27 y^{3} - 216 y^{2}) + 72 y' (576 y)$ .

$72 y' (27 y^{3} - 216 y^{2} + 576 y) + 72 y' \cdot - 512 = 1$

Step 5.2.7

Factor $72 y'$ out of $72 y' (27 y^{3} - 216 y^{2} + 576 y) + 72 y' \cdot - 512$ .

$72 y' (27 y^{3} - 216 y^{2} + 576 y - 512) = 1$

Step 5.3

Divide each term in $72 y' (27 y^{3} - 216 y^{2} + 576 y - 512) = 1$ by $72 (27 y^{3} - 216 y^{2} + 576 y - 512)$ and simplify.

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Step 5.3.1

Divide each term in $72 y' (27 y^{3} - 216 y^{2} + 576 y - 512) = 1$ by $72 (27 y^{3} - 216 y^{2} + 576 y - 512)$ .

$\frac{72 y' (27 y^{3} - 216 y^{2} + 576 y - 512)}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)} = \frac{1}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $72$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{72 y' (27 y^{3} - 216 y^{2} + 576 y - 512)}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)} = \frac{1}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (27 y^{3} - 216 y^{2} + 576 y - 512)}{27 y^{3} - 216 y^{2} + 576 y - 512} = \frac{1}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)}$

Step 5.3.2.2

Cancel the common factor of $27 y^{3} - 216 y^{2} + 576 y - 512$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (27 y^{3} - 216 y^{2} + 576 y - 512)}{27 y^{3} - 216 y^{2} + 576 y - 512} = \frac{1}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{72 (27 y^{3} - 216 y^{2} + 576 y - 512)}$