Calculus Examples

$e^{{(1 + 5 t)}^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = {(1 + 5 t)}^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(1 + 5 t)}^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d t} [{(1 + 5 t)}^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d t} [{(1 + 5 t)}^{2}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(1 + 5 t)}^{2}$ .

$e^{{(1 + 5 t)}^{2}} \frac{d}{d t} [{(1 + 5 t)}^{2}]$

Step 2

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 $g (t) = 1 + 5 t$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $1 + 5 t$ .

$e^{{(1 + 5 t)}^{2}} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d t} [1 + 5 t])$

Step 2.2

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

$e^{{(1 + 5 t)}^{2}} (2 u_{2} \frac{d}{d t} [1 + 5 t])$

Step 2.3

Replace all occurrences of $u_{2}$ with $1 + 5 t$ .

$e^{{(1 + 5 t)}^{2}} (2 (1 + 5 t) \frac{d}{d t} [1 + 5 t])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 + 5 t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [5 t]$ .

$e^{{(1 + 5 t)}^{2}} (2 (1 + 5 t) (\frac{d}{d t} [1] + \frac{d}{d t} [5 t]))$

Step 3.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$e^{{(1 + 5 t)}^{2}} (2 (1 + 5 t) (0 + \frac{d}{d t} [5 t]))$

Step 3.3

Add $0$ and $\frac{d}{d t} [5 t]$ .

$e^{{(1 + 5 t)}^{2}} (2 (1 + 5 t) \frac{d}{d t} [5 t])$

Step 3.4

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

$e^{{(1 + 5 t)}^{2}} (2 (1 + 5 t) (5 \frac{d}{d t} [t]))$

Step 3.5

Multiply $5$ by $2$ .

$e^{{(1 + 5 t)}^{2}} (10 (1 + 5 t) \frac{d}{d t} [t])$

Step 3.6

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

$e^{{(1 + 5 t)}^{2}} (10 (1 + 5 t) \cdot 1)$

Step 3.7

Multiply $10$ by $1$ .

$e^{{(1 + 5 t)}^{2}} (10 (1 + 5 t))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$e^{{(1 + 5 t)}^{2}} (10 \cdot 1 + 10 (5 t))$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $10$ by $1$ .

$e^{{(1 + 5 t)}^{2}} (10 + 10 (5 t))$

Step 4.2.2

Multiply $5$ by $10$ .

$e^{{(1 + 5 t)}^{2}} (10 + 50 t)$