Calculus Examples

${(e^{t^{2}} + 5 t)}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{3}$ and $g (t) = e^{t^{2}} + 5 t$ .

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Step 1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d t} [e^{t^{2}} + 5 t]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$3 {u_{1}}^{2} \frac{d}{d t} [e^{t^{2}} + 5 t]$

Step 1.3

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

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

Step 2

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

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $t^{2}$ .

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

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3

Replace all occurrences of $u_{2}$ with $t^{2}$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$3 {(e^{t^{2}} + 5 t)}^{2} (e^{t^{2}} (2 t) + 5 \cdot 1)$

Step 4.4

Multiply $5$ by $1$ .

$3 {(e^{t^{2}} + 5 t)}^{2} (e^{t^{2}} (2 t) + 5)$