Calculus Examples

$f' (1) = - x^{2} + 9 (x) - 7$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- x^{2} + 9 (x) - 7$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9 (x)] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9 (x)] + \frac{d}{d x} [- 7]$

Step 1.2

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 1.2.1

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

$- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 (x)] + \frac{d}{d x} [- 7]$

Step 1.2.2

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

$- (2 x) + \frac{d}{d x} [9 (x)] + \frac{d}{d x} [- 7]$

Step 1.2.3

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [9 (x)] + \frac{d}{d x} [- 7]$

Step 1.3

Evaluate $\frac{d}{d x} [9 (x)]$ .

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Step 1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$- 2 x + 9 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]$

Step 1.3.2

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

$- 2 x + 9 \cdot 1 + \frac{d}{d x} [- 7]$

Step 1.3.3

Multiply $9$ by $1$ .

$- 2 x + 9 + \frac{d}{d x} [- 7]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$- 2 x + 9 + 0$

Step 1.4.2

Add $- 2 x + 9$ and $0$ .

$f' (x) = - 2 x + 9$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 2 x + 9$ with respect to $x$ is $\frac{d}{d x} [- 2 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [- 2 x] + \frac{d}{d x} [9]$

Step 2.2

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.2.1

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

$- 2 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

Step 2.2.2

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

$- 2 \cdot 1 + \frac{d}{d x} [9]$

Step 2.2.3

Multiply $- 2$ by $1$ .

$- 2 + \frac{d}{d x} [9]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- 2 + 0$

Step 2.3.2

Add $- 2$ and $0$ .

$f'' (x) = - 2$

Step 3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f''' (x) = 0$

Step 4

Since $0$ is constant with respect to $x$ , the derivative of $0$ with respect to $x$ is $0$ .

$f^{4} (x) = 0$