Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Since $- 5$ is constant with respect to $t$ , the derivative of $- 5$ with respect to $t$ is $0$ .

$2 (t^{2} - 5) (2 t + 0)$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 t$ and $0$ .

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

Step 2.4.2

Multiply $2$ by $2$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

$4 t^{2} t + 4 \cdot - 5 t$

Step 3.3

Combine terms.

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Step 3.3.1

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

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

Step 3.3.2

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

$4 t^{1 + 2} + 4 \cdot - 5 t$

Step 3.3.3

Add $1$ and $2$ .

$4 t^{3} + 4 \cdot - 5 t$

Step 3.3.4

Multiply $4$ by $- 5$ .

$4 t^{3} - 20 t$