Calculus Examples

$\frac{d^{2}}{d x^{2}} (\sin (x) + \cos (x))$

Step 1

Find the first derivative.

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Step 1.1

Cancel the common factor of $d^{2}$ and $d$ .

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Step 1.1.1

Factor $d$ out of $d^{2}$ .

$\frac{d}{d \cdot d} [\frac{d \cdot d}{d x^{2}} (\sin (x) + \cos (x))]$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $d$ out of $d x^{2}$ .

$\frac{d}{d \cdot d} [\frac{d \cdot d}{d (x^{2})} (\sin (x) + \cos (x))]$

Step 1.1.2.2

Cancel the common factor.

$\frac{d}{d \cdot d} [\frac{d \cdot d}{d x^{2}} (\sin (x) + \cos (x))]$

Step 1.1.2.3

Rewrite the expression.

$\frac{d}{d \cdot d} [\frac{d}{x^{2}} (\sin (x) + \cos (x))]$

Step 1.2

Since $\sin (x) + \cos (x)$ is constant with respect to $d$ , the derivative of $\frac{d}{x^{2}} (\sin (x) + \cos (x))$ with respect to $d$ is $(\sin (x) + \cos (x)) \frac{d}{d \cdot d} [\frac{d}{x^{2}}]$ .

$(\sin (x) + \cos (x)) \frac{d}{d \cdot d} [\frac{d}{x^{2}}]$

Step 1.3

Since $\frac{1}{x^{2}}$ is constant with respect to $d$ , the derivative of $\frac{d}{x^{2}}$ with respect to $d$ is $\frac{1}{x^{2}} \frac{d}{d \cdot d} [d]$ .

$(\sin (x) + \cos (x)) (\frac{1}{x^{2}} \frac{d}{d \cdot d} [d])$

Step 1.4

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

$(\sin (x) + \cos (x)) \frac{1}{x^{2}} \cdot 1$

Step 1.5

Multiply $\sin (x) + \cos (x)$ by $1$ .

$(\sin (x) + \cos (x)) \frac{1}{x^{2}}$

Step 1.6

Simplify.

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Step 1.6.1

Apply the distributive property.

$\sin (x) \frac{1}{x^{2}} + \cos (x) \frac{1}{x^{2}}$

Step 1.6.2

Combine terms.

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Step 1.6.2.1

Combine $\sin (x)$ and $\frac{1}{x^{2}}$ .

$\frac{\sin (x)}{x^{2}} + \cos (x) \frac{1}{x^{2}}$

Step 1.6.2.2

Combine $\cos (x)$ and $\frac{1}{x^{2}}$ .

$f' (d) = \frac{\sin (x)}{x^{2}} + \frac{\cos (x)}{x^{2}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{\sin (x)}{x^{2}} + \frac{\cos (x)}{x^{2}}$ with respect to $d$ is $\frac{d}{d \cdot d} [\frac{\sin (x)}{x^{2}}] + \frac{d}{d \cdot d} [\frac{\cos (x)}{x^{2}}]$ .

$\frac{d}{d \cdot d} [\frac{\sin (x)}{x^{2}}] + \frac{d}{d \cdot d} [\frac{\cos (x)}{x^{2}}]$

Step 2.2

Since $\frac{\sin (x)}{x^{2}}$ is constant with respect to $d$ , the derivative of $\frac{\sin (x)}{x^{2}}$ with respect to $d$ is $0$ .

$0 + \frac{d}{d \cdot d} [\frac{\cos (x)}{x^{2}}]$

Step 2.3

Since $\frac{\cos (x)}{x^{2}}$ is constant with respect to $d$ , the derivative of $\frac{\cos (x)}{x^{2}}$ with respect to $d$ is $0$ .

$0 + 0$

Step 2.4

Add $0$ and $0$ .

$f'' (d) = 0$