Calculus Examples

${(8 t - 7)}^{4} + {(8 t + 7)}^{4}$

Step 1

By the Sum Rule, the derivative of ${(8 t - 7)}^{4} + {(8 t + 7)}^{4}$ with respect to $t$ is $\frac{d}{d t} [{(8 t - 7)}^{4}] + \frac{d}{d t} [{(8 t + 7)}^{4}]$ .

$\frac{d}{d t} [{(8 t - 7)}^{4}] + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2

Evaluate $\frac{d}{d t} [{(8 t - 7)}^{4}]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{4}$ and $g (t) = 8 t - 7$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $8 t - 7$ .

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d t} [8 t - 7] + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.1.2

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

$4 {u_{1}}^{3} \frac{d}{d t} [8 t - 7] + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $8 t - 7$ .

$4 {(8 t - 7)}^{3} \frac{d}{d t} [8 t - 7] + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.2

By the Sum Rule, the derivative of $8 t - 7$ with respect to $t$ is $\frac{d}{d t} [8 t] + \frac{d}{d t} [- 7]$ .

$4 {(8 t - 7)}^{3} (\frac{d}{d t} [8 t] + \frac{d}{d t} [- 7]) + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.3

Since $8$ is constant with respect to $t$ , the derivative of $8 t$ with respect to $t$ is $8 \frac{d}{d t} [t]$ .

$4 {(8 t - 7)}^{3} (8 \frac{d}{d t} [t] + \frac{d}{d t} [- 7]) + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.4

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

$4 {(8 t - 7)}^{3} (8 \cdot 1 + \frac{d}{d t} [- 7]) + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.5

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7$ with respect to $t$ is $0$ .

$4 {(8 t - 7)}^{3} (8 \cdot 1 + 0) + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.6

Multiply $8$ by $1$ .

$4 {(8 t - 7)}^{3} (8 + 0) + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.7

Add $8$ and $0$ .

$4 {(8 t - 7)}^{3} \cdot 8 + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 2.8

Multiply $8$ by $4$ .

$32 {(8 t - 7)}^{3} + \frac{d}{d t} [{(8 t + 7)}^{4}]$

Step 3

Evaluate $\frac{d}{d t} [{(8 t + 7)}^{4}]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{4}$ and $g (t) = 8 t + 7$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $8 t + 7$ .

$32 {(8 t - 7)}^{3} + \frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d t} [8 t + 7]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$32 {(8 t - 7)}^{3} + 4 {u_{2}}^{3} \frac{d}{d t} [8 t + 7]$

Step 3.1.3

Replace all occurrences of $u_{2}$ with $8 t + 7$ .

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} \frac{d}{d t} [8 t + 7]$

Step 3.2

By the Sum Rule, the derivative of $8 t + 7$ with respect to $t$ is $\frac{d}{d t} [8 t] + \frac{d}{d t} [7]$ .

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} (\frac{d}{d t} [8 t] + \frac{d}{d t} [7])$

Step 3.3

Since $8$ is constant with respect to $t$ , the derivative of $8 t$ with respect to $t$ is $8 \frac{d}{d t} [t]$ .

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} (8 \frac{d}{d t} [t] + \frac{d}{d t} [7])$

Step 3.4

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

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} (8 \cdot 1 + \frac{d}{d t} [7])$

Step 3.5

Since $7$ is constant with respect to $t$ , the derivative of $7$ with respect to $t$ is $0$ .

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} (8 \cdot 1 + 0)$

Step 3.6

Multiply $8$ by $1$ .

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} (8 + 0)$

Step 3.7

Add $8$ and $0$ .

$32 {(8 t - 7)}^{3} + 4 {(8 t + 7)}^{3} \cdot 8$

Step 3.8

Multiply $8$ by $4$ .

$32 {(8 t - 7)}^{3} + 32 {(8 t + 7)}^{3}$

Step 4

Simplify.

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Step 4.1

Factor $32$ out of $32 {(8 t - 7)}^{3} + 32 {(8 t + 7)}^{3}$ .

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Step 4.1.1

Factor $32$ out of $32 {(8 t - 7)}^{3}$ .

$32 ({(8 t - 7)}^{3}) + 32 {(8 t + 7)}^{3}$

Step 4.1.2

Factor $32$ out of $32 {(8 t + 7)}^{3}$ .

$32 ({(8 t - 7)}^{3}) + 32 ({(8 t + 7)}^{3})$

Step 4.1.3

Factor $32$ out of $32 ({(8 t - 7)}^{3}) + 32 ({(8 t + 7)}^{3})$ .

$32 ({(8 t - 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2

Simplify each term.

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Step 4.2.1

Use the Binomial Theorem.

$32 ({(8 t)}^{3} + 3 {(8 t)}^{2} \cdot - 7 + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Apply the product rule to $8 t$ .

$32 (8^{3} t^{3} + 3 {(8 t)}^{2} \cdot - 7 + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.2

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

$32 (512 t^{3} + 3 {(8 t)}^{2} \cdot - 7 + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.3

Apply the product rule to $8 t$ .

$32 (512 t^{3} + 3 (8^{2} t^{2}) \cdot - 7 + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.4

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

$32 (512 t^{3} + 3 (64 t^{2}) \cdot - 7 + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.5

Multiply $64$ by $3$ .

$32 (512 t^{3} + 192 t^{2} \cdot - 7 + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.6

Multiply $- 7$ by $192$ .

$32 (512 t^{3} - 1344 t^{2} + 3 (8 t) {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.7

Multiply $8$ by $3$ .

$32 (512 t^{3} - 1344 t^{2} + 24 t {(- 7)}^{2} + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.8

Raise $- 7$ to the power of $2$ .

$32 (512 t^{3} - 1344 t^{2} + 24 t \cdot 49 + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.9

Multiply $49$ by $24$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t + {(- 7)}^{3} + {(8 t + 7)}^{3})$

Step 4.2.2.10

Raise $- 7$ to the power of $3$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + {(8 t + 7)}^{3})$

Step 4.2.3

Use the Binomial Theorem.

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + {(8 t)}^{3} + 3 {(8 t)}^{2} \cdot 7 + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Apply the product rule to $8 t$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 8^{3} t^{3} + 3 {(8 t)}^{2} \cdot 7 + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4.2

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

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 3 {(8 t)}^{2} \cdot 7 + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4.3

Apply the product rule to $8 t$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 3 (8^{2} t^{2}) \cdot 7 + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4.4

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

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 3 (64 t^{2}) \cdot 7 + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4.5

Multiply $64$ by $3$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 192 t^{2} \cdot 7 + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4.6

Multiply $7$ by $192$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 1344 t^{2} + 3 (8 t) \cdot 7^{2} + 7^{3})$

Step 4.2.4.7

Multiply $8$ by $3$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 1344 t^{2} + 24 t \cdot 7^{2} + 7^{3})$

Step 4.2.4.8

Raise $7$ to the power of $2$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 1344 t^{2} + 24 t \cdot 49 + 7^{3})$

Step 4.2.4.9

Multiply $49$ by $24$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 1344 t^{2} + 1176 t + 7^{3})$

Step 4.2.4.10

Raise $7$ to the power of $3$ .

$32 (512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 1344 t^{2} + 1176 t + 343)$

Step 4.3

Combine the opposite terms in $512 t^{3} - 1344 t^{2} + 1176 t - 343 + 512 t^{3} + 1344 t^{2} + 1176 t + 343$ .

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Step 4.3.1

Add $- 1344 t^{2}$ and $1344 t^{2}$ .

$32 (512 t^{3} + 1176 t - 343 + 512 t^{3} + 0 + 1176 t + 343)$

Step 4.3.2

Add $512 t^{3} + 1176 t - 343 + 512 t^{3}$ and $0$ .

$32 (512 t^{3} + 1176 t - 343 + 512 t^{3} + 1176 t + 343)$

Step 4.3.3

Add $- 343$ and $343$ .

$32 (512 t^{3} + 1176 t + 512 t^{3} + 1176 t + 0)$

Step 4.3.4

Add $512 t^{3} + 1176 t + 512 t^{3} + 1176 t$ and $0$ .

$32 (512 t^{3} + 1176 t + 512 t^{3} + 1176 t)$

Step 4.4

Add $512 t^{3}$ and $512 t^{3}$ .

$32 (1024 t^{3} + 1176 t + 1176 t)$

Step 4.5

Add $1176 t$ and $1176 t$ .

$32 (1024 t^{3} + 2352 t)$

Step 4.6

Apply the distributive property.

$32 (1024 t^{3}) + 32 (2352 t)$

Step 4.7

Multiply $1024$ by $32$ .

$32768 t^{3} + 32 (2352 t)$

Step 4.8

Multiply $2352$ by $32$ .

$32768 t^{3} + 75264 t$