Calculus Examples

y = 3 t {(2 t^{3} - 5)}^{5}

$y = 3 t {(2 t^{3} - 5)}^{5}$

Step 1

Differentiate both sides of the equation.

\frac{d}{d t} (y) = \frac{d}{d t} (3 t {(2 t^{3} - 5)}^{5})

$\frac{d}{d t} (y) = \frac{d}{d t} (3 t {(2 t^{3} - 5)}^{5})$

Step 2

The derivative of

y

$y$ with respect to

t

$t$ is

$y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since

$3$ is constant with respect to

$t$ , the derivative of

$3 t {(2 t^{3} - 5)}^{5}$ with respect to

$t$ is

$3 \frac{d}{d t} [t {(2 t^{3} - 5)}^{5}]$ .

$3 \frac{d}{d t} [t {(2 t^{3} - 5)}^{5}]$

Step 3.2

Differentiate using the Product Rule which states that

$\frac{d}{d t} [f (t) g (t)]$ is

$f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where

$f (t) = t$ and

$g (t) = {(2 t^{3} - 5)}^{5}$ .

$3 (t \frac{d}{d t} [{(2 t^{3} - 5)}^{5}] + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.3

Differentiate using the chain rule, which states that

$\frac{d}{d t} [f (g (t))]$ is

$f' (g (t)) g' (t)$ where

$f (t) = t^{5}$ and

$g (t) = 2 t^{3} - 5$ .

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Step 3.3.1

To apply the Chain Rule, set

$u$ as

$2 t^{3} - 5$ .

$3 (t (\frac{d}{d u} [u^{5}] \frac{d}{d t} [2 t^{3} - 5]) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 5$ .

$3 (t (5 u^{4} \frac{d}{d t} [2 t^{3} - 5]) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.3.3

Replace all occurrences of

$u$ with

$2 t^{3} - 5$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} \frac{d}{d t} [2 t^{3} - 5]) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of

$2 t^{3} - 5$ with respect to

$t$ is

$\frac{d}{d t} [2 t^{3}] + \frac{d}{d t} [- 5]$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} (\frac{d}{d t} [2 t^{3}] + \frac{d}{d t} [- 5])) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4.2

Since

$2$ is constant with respect to

$t$ , the derivative of

$2 t^{3}$ with respect to

$t$ is

$2 \frac{d}{d t} [t^{3}]$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} (2 \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 5])) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4.3

Differentiate using the Power Rule which states that

$\frac{d}{d t} [t^{n}]$ is

$n t^{n - 1}$ where

$n = 3$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} (2 (3 t^{2}) + \frac{d}{d t} [- 5])) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4.4

Multiply

$3$ by

$2$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} (6 t^{2} + \frac{d}{d t} [- 5])) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4.5

Since

$- 5$ is constant with respect to

$t$ , the derivative of

$- 5$ with respect to

$t$ is

$0$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} (6 t^{2} + 0)) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Add

$6 t^{2}$ and

$0$ .

$3 (t (5 {(2 t^{3} - 5)}^{4} (6 t^{2})) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.4.6.2

Multiply

$6$ by

$5$ .

$3 (t (30 {(2 t^{3} - 5)}^{4} t^{2}) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.5

Raise

$t$ to the power of

$1$ .

$3 (t^{2} t^{1} (30 {(2 t^{3} - 5)}^{4}) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.6

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 (t^{2 + 1} (30 {(2 t^{3} - 5)}^{4}) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.7

Add

$2$ and

$1$ .

$3 (t^{3} (30 {(2 t^{3} - 5)}^{4}) + {(2 t^{3} - 5)}^{5} \frac{d}{d t} [t])$

Step 3.8

Differentiate using the Power Rule which states that

$\frac{d}{d t} [t^{n}]$ is

$n t^{n - 1}$ where

$n = 1$ .

$3 (t^{3} (30 {(2 t^{3} - 5)}^{4}) + {(2 t^{3} - 5)}^{5} \cdot 1)$

Step 3.9

Multiply

${(2 t^{3} - 5)}^{5}$ by

$1$ .

$3 (t^{3} (30 {(2 t^{3} - 5)}^{4}) + {(2 t^{3} - 5)}^{5})$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$3 (t^{3} (30 {(2 t^{3} - 5)}^{4})) + 3 {(2 t^{3} - 5)}^{5}$

Step 3.10.2

Multiply

$30$ by

$3$ .

$90 (t^{3} ({(2 t^{3} - 5)}^{4})) + 3 {(2 t^{3} - 5)}^{5}$

Step 3.10.3

Factor

$3 {(2 t^{3} - 5)}^{4}$ out of

$90 t^{3} {(2 t^{3} - 5)}^{4} + 3 {(2 t^{3} - 5)}^{5}$ .

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Step 3.10.3.1

Factor

$3 {(2 t^{3} - 5)}^{4}$ out of

$90 t^{3} {(2 t^{3} - 5)}^{4}$ .

$3 {(2 t^{3} - 5)}^{4} (30 t^{3}) + 3 {(2 t^{3} - 5)}^{5}$

Step 3.10.3.2

Factor

$3 {(2 t^{3} - 5)}^{4}$ out of

$3 {(2 t^{3} - 5)}^{5}$ .

$3 {(2 t^{3} - 5)}^{4} (30 t^{3}) + 3 {(2 t^{3} - 5)}^{4} (2 t^{3} - 5)$

Step 3.10.3.3

Factor

$3 {(2 t^{3} - 5)}^{4}$ out of

$3 {(2 t^{3} - 5)}^{4} (30 t^{3}) + 3 {(2 t^{3} - 5)}^{4} (2 t^{3} - 5)$ .

$3 {(2 t^{3} - 5)}^{4} (30 t^{3} + 2 t^{3} - 5)$

Step 3.10.4

Add

$30 t^{3}$ and

$2 t^{3}$ .

$3 {(2 t^{3} - 5)}^{4} (32 t^{3} - 5)$

Step 4

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

$y' = 3 {(2 t^{3} - 5)}^{4} (32 t^{3} - 5)$

Step 5

Replace

$y'$ with

$\frac{d y}{d t}$ .

$\frac{d y}{d t} = 3 {(2 t^{3} - 5)}^{4} (32 t^{3} - 5)$