Calculus Examples

$y = {(5 t^{2} - 4 t)}^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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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 $f (t) = t^{2}$ and $g (t) = 5 t^{2} - 4 t$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $5 t^{2} - 4 t$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d t} [5 t^{2} - 4 t]$

Step 3.1.2

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

$2 u \frac{d}{d t} [5 t^{2} - 4 t]$

Step 3.1.3

Replace all occurrences of $u$ with $5 t^{2} - 4 t$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $5 t^{2} - 4 t$ with respect to $t$ is $\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [- 4 t]$ .

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

Step 3.2.2

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{2}$ with respect to $t$ is $5 \frac{d}{d t} [t^{2}]$ .

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $2$ by $5$ .

$2 (5 t^{2} - 4 t) (10 t + \frac{d}{d t} [- 4 t])$

Step 3.2.5

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4 t$ with respect to $t$ is $- 4 \frac{d}{d t} [t]$ .

$2 (5 t^{2} - 4 t) (10 t - 4 \frac{d}{d t} [t])$

Step 3.2.6

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

$2 (5 t^{2} - 4 t) (10 t - 4 \cdot 1)$

Step 3.2.7

Multiply $- 4$ by $1$ .

$2 (5 t^{2} - 4 t) (10 t - 4)$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$(2 (5 t^{2}) + 2 (- 4 t)) (10 t - 4)$

Step 3.3.2

Combine terms.

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Step 3.3.2.1

Multiply $5$ by $2$ .

$(10 t^{2} + 2 (- 4 t)) (10 t - 4)$

Step 3.3.2.2

Multiply $- 4$ by $2$ .

$(10 t^{2} - 8 t) (10 t - 4)$

Step 3.3.3

Reorder the factors of $(10 t^{2} - 8 t) (10 t - 4)$ .

$(10 t - 4) (10 t^{2} - 8 t)$

Step 3.3.4

Expand $(10 t - 4) (10 t^{2} - 8 t)$ using the FOIL Method.

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Step 3.3.4.1

Apply the distributive property.

$10 t (10 t^{2} - 8 t) - 4 (10 t^{2} - 8 t)$

Step 3.3.4.2

Apply the distributive property.

$10 t (10 t^{2}) + 10 t (- 8 t) - 4 (10 t^{2} - 8 t)$

Step 3.3.4.3

Apply the distributive property.

$10 t (10 t^{2}) + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5

Simplify and combine like terms.

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Step 3.3.5.1

Simplify each term.

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Step 3.3.5.1.1

Rewrite using the commutative property of multiplication.

$10 \cdot 10 t \cdot t^{2} + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.2

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 3.3.5.1.2.1

Move $t^{2}$ .

$10 \cdot 10 (t^{2} t) + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.2.2

Multiply $t^{2}$ by $t$ .

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Step 3.3.5.1.2.2.1

Raise $t$ to the power of $1$ .

$10 \cdot 10 (t^{2} t^{1}) + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 \cdot 10 t^{2 + 1} + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.2.3

Add $2$ and $1$ .

$10 \cdot 10 t^{3} + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.3

Multiply $10$ by $10$ .

$100 t^{3} + 10 t (- 8 t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.4

Rewrite using the commutative property of multiplication.

$100 t^{3} + 10 \cdot - 8 t \cdot t - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.5

Multiply $t$ by $t$ by adding the exponents.

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Step 3.3.5.1.5.1

Move $t$ .

$100 t^{3} + 10 \cdot - 8 (t \cdot t) - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.5.2

Multiply $t$ by $t$ .

$100 t^{3} + 10 \cdot - 8 t^{2} - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.6

Multiply $10$ by $- 8$ .

$100 t^{3} - 80 t^{2} - 4 (10 t^{2}) - 4 (- 8 t)$

Step 3.3.5.1.7

Multiply $10$ by $- 4$ .

$100 t^{3} - 80 t^{2} - 40 t^{2} - 4 (- 8 t)$

Step 3.3.5.1.8

Multiply $- 8$ by $- 4$ .

$100 t^{3} - 80 t^{2} - 40 t^{2} + 32 t$

Step 3.3.5.2

Subtract $40 t^{2}$ from $- 80 t^{2}$ .

$100 t^{3} - 120 t^{2} + 32 t$

Step 4

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

$y' = 100 t^{3} - 120 t^{2} + 32 t$

Step 5

Replace $y'$ with $\frac{d y}{d t}$ .

$\frac{d y}{d t} = 100 t^{3} - 120 t^{2} + 32 t$