Calculus Examples

$y = {(e^{\cos (\frac{t}{7})})}^{4}$

Step 1

Multiply the exponents in ${(e^{\cos (\frac{t}{7})})}^{4}$ .

Tap for more steps...

Step 1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = e^{\cos (\frac{t}{7}) \cdot 4}$

Step 1.2

Move $4$ to the left of $\cos (\frac{t}{7})$ .

$y = e^{4 \cos (\frac{t}{7})}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (e^{4 \cos (\frac{t}{7})})$

Step 3

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

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

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 $g (t) = 4 \cos (\frac{t}{7})$ .

Tap for more steps...

Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $4 \cos (\frac{t}{7})$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d t} [4 \cos (\frac{t}{7})]$

Step 4.1.2

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

$e^{u_{1}} \frac{d}{d t} [4 \cos (\frac{t}{7})]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $4 \cos (\frac{t}{7})$ .

$e^{4 \cos (\frac{t}{7})} \frac{d}{d t} [4 \cos (\frac{t}{7})]$

Step 4.2

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 4.2.1

Since $4$ is constant with respect to $t$ , the derivative of $4 \cos (\frac{t}{7})$ with respect to $t$ is $4 \frac{d}{d t} [\cos (\frac{t}{7})]$ .

$e^{4 \cos (\frac{t}{7})} (4 \frac{d}{d t} [\cos (\frac{t}{7})])$

Step 4.2.2

Move $4$ to the left of $e^{4 \cos (\frac{t}{7})}$ .

$4 \cdot e^{4 \cos (\frac{t}{7})} \frac{d}{d t} [\cos (\frac{t}{7})]$

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = \frac{t}{7}$ .

Tap for more steps...

Step 4.3.1

To apply the Chain Rule, set $u_{2}$ as $\frac{t}{7}$ .

$4 e^{4 \cos (\frac{t}{7})} (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d t} [\frac{t}{7}])$

Step 4.3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$4 e^{4 \cos (\frac{t}{7})} (- \sin (u_{2}) \frac{d}{d t} [\frac{t}{7}])$

Step 4.3.3

Replace all occurrences of $u_{2}$ with $\frac{t}{7}$ .

$4 e^{4 \cos (\frac{t}{7})} (- \sin (\frac{t}{7}) \frac{d}{d t} [\frac{t}{7}])$

Step 4.4

Differentiate.

Tap for more steps...

Step 4.4.1

Multiply $- 1$ by $4$ .

$- 4 e^{4 \cos (\frac{t}{7})} (\sin (\frac{t}{7}) \frac{d}{d t} [\frac{t}{7}])$

Step 4.4.2

Since $\frac{1}{7}$ is constant with respect to $t$ , the derivative of $\frac{t}{7}$ with respect to $t$ is $\frac{1}{7} \frac{d}{d t} [t]$ .

$- 4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7}) (\frac{1}{7} \frac{d}{d t} [t])$

Step 4.4.3

Combine fractions.

Tap for more steps...

Step 4.4.3.1

Combine $\frac{1}{7}$ and $- 4$ .

$\frac{- 4}{7} e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7}) \frac{d}{d t} [t]$

Step 4.4.3.2

Combine $\frac{- 4}{7}$ and $e^{4 \cos (\frac{t}{7})}$ .

$\frac{- 4 e^{4 \cos (\frac{t}{7})}}{7} \sin (\frac{t}{7}) \frac{d}{d t} [t]$

Step 4.4.3.3

Combine $\frac{- 4 e^{4 \cos (\frac{t}{7})}}{7}$ and $\sin (\frac{t}{7})$ .

$\frac{- 4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7})}{7} \frac{d}{d t} [t]$

Step 4.4.3.4

Move the negative in front of the fraction.

$- \frac{4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7})}{7} \frac{d}{d t} [t]$

Step 4.4.4

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

$- \frac{4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7})}{7} \cdot 1$

Step 4.4.5

Multiply $- 1$ by $1$ .

$- \frac{4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7})}{7}$

Step 5

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

$y' = - \frac{4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7})}{7}$

Step 6

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

$\frac{d y}{d t} = - \frac{4 e^{4 \cos (\frac{t}{7})} \sin (\frac{t}{7})}{7}$