Calculus Examples

$3 t^{1.8} - \cot (0.8 t)$

Step 1

By the Sum Rule, the derivative of $3 t^{1.8} - \cot (0.8 t)$ with respect to $t$ is $\frac{d}{d t} [3 t^{1.8}] + \frac{d}{d t} [- \cot (0.8 t)]$ .

$\frac{d}{d t} [3 t^{1.8}] + \frac{d}{d t} [- \cot (0.8 t)]$

Step 2

Evaluate $\frac{d}{d t} [3 t^{1.8}]$ .

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Step 2.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{1.8}$ with respect to $t$ is $3 \frac{d}{d t} [t^{1.8}]$ .

$3 \frac{d}{d t} [t^{1.8}] + \frac{d}{d t} [- \cot (0.8 t)]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1.8$ .

$3 (1.8 t^{0.8}) + \frac{d}{d t} [- \cot (0.8 t)]$

Step 2.3

Multiply $1.8$ by $3$ .

$5.4 t^{0.8} + \frac{d}{d t} [- \cot (0.8 t)]$

Step 3

Evaluate $\frac{d}{d t} [- \cot (0.8 t)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- \cot (0.8 t)$ with respect to $t$ is $- \frac{d}{d t} [\cot (0.8 t)]$ .

$5.4 t^{0.8} - \frac{d}{d t} [\cot (0.8 t)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cot (t)$ and $g (t) = 0.8 t$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $0.8 t$ .

$5.4 t^{0.8} - (\frac{d}{d u} [\cot (u)] \frac{d}{d t} [0.8 t])$

Step 3.2.2

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

$5.4 t^{0.8} - (- \csc^{2} (u) \frac{d}{d t} [0.8 t])$

Step 3.2.3

Replace all occurrences of $u$ with $0.8 t$ .

$5.4 t^{0.8} - (- \csc^{2} (0.8 t) \frac{d}{d t} [0.8 t])$

Step 3.3

Since $0.8$ is constant with respect to $t$ , the derivative of $0.8 t$ with respect to $t$ is $0.8 \frac{d}{d t} [t]$ .

$5.4 t^{0.8} - (- \csc^{2} (0.8 t) (0.8 \frac{d}{d t} [t]))$

Step 3.4

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

$5.4 t^{0.8} - (- \csc^{2} (0.8 t) (0.8 \cdot 1))$

Step 3.5

Multiply $0.8$ by $1$ .

$5.4 t^{0.8} - (- \csc^{2} (0.8 t) \cdot 0.8)$

Step 3.6

Multiply $0.8$ by $- 1$ .

$5.4 t^{0.8} - (- 0.8 \csc^{2} (0.8 t))$

Step 3.7

Multiply $- 0.8$ by $- 1$ .

$5.4 t^{0.8} + 0.8 \csc^{2} (0.8 t)$