Calculus Examples

$y = e^{\cot (3 z)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = e^{z}$ and $g (z) = \cot (3 z)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cot (3 z)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d z} [\cot (3 z)]$

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 $a$ = $e$ .

$e^{u_{1}} \frac{d}{d z} [\cot (3 z)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cot (3 z)$ .

$e^{\cot (3 z)} \frac{d}{d z} [\cot (3 z)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = \cot (z)$ and $g (z) = 3 z$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $3 z$ .

$e^{\cot (3 z)} (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d z} [3 z])$

Step 2.2

The derivative of $\cot (u_{2})$ with respect to $u_{2}$ is $- \csc^{2} (u_{2})$ .

$e^{\cot (3 z)} (- \csc^{2} (u_{2}) \frac{d}{d z} [3 z])$

Step 2.3

Replace all occurrences of $u_{2}$ with $3 z$ .

$e^{\cot (3 z)} (- \csc^{2} (3 z) \frac{d}{d z} [3 z])$

Step 3

Differentiate.

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Step 3.1

Since $3$ is constant with respect to $z$ , the derivative of $3 z$ with respect to $z$ is $3 \frac{d}{d z} [z]$ .

$e^{\cot (3 z)} (- \csc^{2} (3 z) (3 \frac{d}{d z} [z]))$

Step 3.2

Multiply $3$ by $- 1$ .

$e^{\cot (3 z)} (- 3 \csc^{2} (3 z) \frac{d}{d z} [z])$

Step 3.3

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

$e^{\cot (3 z)} (- 3 \csc^{2} (3 z) \cdot 1)$

Step 3.4

Simplify the expression.

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Step 3.4.1

Multiply $- 3$ by $1$ .

$e^{\cot (3 z)} (- 3 \csc^{2} (3 z))$

Step 3.4.2

Reorder the factors of $e^{\cot (3 z)} (- 3 \csc^{2} (3 z))$ .

$- 3 e^{\cot (3 z)} \csc^{2} (3 z)$