Calculus Examples

$v (r) = k (26 r^{2} - r^{3})$

Step 1

Find the first derivative.

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Step 1.1

Since $k$ is constant with respect to $r$ , the derivative of $k (26 r^{2} - r^{3})$ with respect to $r$ is $k \frac{d}{d r} [26 r^{2} - r^{3}]$ .

$k \frac{d}{d r} [26 r^{2} - r^{3}]$

Step 1.2

By the Sum Rule, the derivative of $26 r^{2} - r^{3}$ with respect to $r$ is $\frac{d}{d r} [26 r^{2}] + \frac{d}{d r} [- r^{3}]$ .

$k (\frac{d}{d r} [26 r^{2}] + \frac{d}{d r} [- r^{3}])$

Step 1.3

Since $26$ is constant with respect to $r$ , the derivative of $26 r^{2}$ with respect to $r$ is $26 \frac{d}{d r} [r^{2}]$ .

$k (26 \frac{d}{d r} [r^{2}] + \frac{d}{d r} [- r^{3}])$

Step 1.4

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

$k (26 (2 r) + \frac{d}{d r} [- r^{3}])$

Step 1.5

Multiply $2$ by $26$ .

$k (52 r + \frac{d}{d r} [- r^{3}])$

Step 1.6

Since $- 1$ is constant with respect to $r$ , the derivative of $- r^{3}$ with respect to $r$ is $- \frac{d}{d r} [r^{3}]$ .

$k (52 r - \frac{d}{d r} [r^{3}])$

Step 1.7

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

$k (52 r - (3 r^{2}))$

Step 1.8

Multiply $3$ by $- 1$ .

$k (52 r - 3 r^{2})$

Step 1.9

Simplify.

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Step 1.9.1

Apply the distributive property.

$k (52 r) + k (- 3 r^{2})$

Step 1.9.2

Move $52$ to the left of $k$ .

$52 k r + k (- 3 r^{2})$

Step 1.9.3

Reorder terms.

$f' (r) = 52 k r - 3 r^{2} k$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $52 k r - 3 r^{2} k$ with respect to $r$ is $\frac{d}{d r} [52 k r] + \frac{d}{d r} [- 3 r^{2} k]$ .

$\frac{d}{d r} [52 k r] + \frac{d}{d r} [- 3 r^{2} k]$

Step 2.2

Evaluate $\frac{d}{d r} [52 k r]$ .

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Step 2.2.1

Since $52 k$ is constant with respect to $r$ , the derivative of $52 k r$ with respect to $r$ is $52 k \frac{d}{d r} [r]$ .

$52 k \frac{d}{d r} [r] + \frac{d}{d r} [- 3 r^{2} k]$

Step 2.2.2

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

$52 k \cdot 1 + \frac{d}{d r} [- 3 r^{2} k]$

Step 2.2.3

Multiply $52$ by $1$ .

$52 k + \frac{d}{d r} [- 3 r^{2} k]$

Step 2.3

Evaluate $\frac{d}{d r} [- 3 r^{2} k]$ .

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Step 2.3.1

Since $- 3 k$ is constant with respect to $r$ , the derivative of $- 3 r^{2} k$ with respect to $r$ is $- 3 k \frac{d}{d r} [r^{2}]$ .

$52 k - 3 k \frac{d}{d r} [r^{2}]$

Step 2.3.2

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

$52 k - 3 k (2 r)$

Step 2.3.3

Multiply $2$ by $- 3$ .

$52 k - 6 k r$

Step 2.4

Reorder terms.

$f'' (r) = - 6 k r + 52 k$

Step 3

The second derivative of $v (r)$ with respect to $r$ is $- 6 k r + 52 k$ .

$- 6 k r + 52 k$