Calculus Examples

$y = - x^{2} + 3$

Step 1

Find the first derivative.

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

$f' (x) = \frac{d}{d x} (- x^{2}) + \frac{d}{d x} (3)$

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

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

$f' (x) = - \frac{d}{d x} x^{2} + \frac{d}{d x} (3)$

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

$f' (x) = - (2 x) + \frac{d}{d x} (3)$

Multiply $2$ by $- 1$ .

$f' (x) = - 2 x + \frac{d}{d x} (3)$

Differentiate using the Constant Rule.

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Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = - 2 x + 0$

Add $- 2 x$ and $0$ .

$f' (x) = - 2 x$

Step 2

Find the second derivative.

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

$f'' (x) = - 2 \frac{d}{d x} x$

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

$f'' (x) = - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$f'' (x) = - 2$

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 $0$ .

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