Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $t$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

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

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

Tap for more steps...

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

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