Calculus Examples

$y = \frac{- 3}{{(3 x^{2} + 1)}^{3}}$ ; $(0, - 3)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = - 3$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{3}{{(3 x^{2} + 1)}^{3}}]$

Step 1.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{{(3 x^{2} + 1)}^{3}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{{(3 x^{2} + 1)}^{3}}]$ .

$- 3 \frac{d}{d x} [\frac{1}{{(3 x^{2} + 1)}^{3}}]$

Step 1.1.3

Apply basic rules of exponents.

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Step 1.1.3.1

Rewrite $\frac{1}{{(3 x^{2} + 1)}^{3}}$ as ${({(3 x^{2} + 1)}^{3})}^{- 1}$ .

$- 3 \frac{d}{d x} [{({(3 x^{2} + 1)}^{3})}^{- 1}]$

Step 1.1.3.2

Multiply the exponents in ${({(3 x^{2} + 1)}^{3})}^{- 1}$ .

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Step 1.1.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 3 \frac{d}{d x} [{(3 x^{2} + 1)}^{3 \cdot - 1}]$

Step 1.1.3.2.2

Multiply $3$ by $- 1$ .

$- 3 \frac{d}{d x} [{(3 x^{2} + 1)}^{- 3}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = 3 x^{2} + 1$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $3 x^{2} + 1$ .

$- 3 (\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [3 x^{2} + 1])$

Step 1.2.2

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

$- 3 (- 3 u^{- 4} \frac{d}{d x} [3 x^{2} + 1])$

Step 1.2.3

Replace all occurrences of $u$ with $3 x^{2} + 1$ .

$- 3 (- 3 {(3 x^{2} + 1)}^{- 4} \frac{d}{d x} [3 x^{2} + 1])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 3$ by $- 3$ .

$9 ({(3 x^{2} + 1)}^{- 4} \frac{d}{d x} [3 x^{2} + 1])$

Step 1.3.2

By the Sum Rule, the derivative of $3 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [1]$ .

$9 {(3 x^{2} + 1)}^{- 4} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [1])$

Step 1.3.3

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

$9 {(3 x^{2} + 1)}^{- 4} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])$

Step 1.3.4

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

$9 {(3 x^{2} + 1)}^{- 4} (3 (2 x) + \frac{d}{d x} [1])$

Step 1.3.5

Multiply $2$ by $3$ .

$9 {(3 x^{2} + 1)}^{- 4} (6 x + \frac{d}{d x} [1])$

Step 1.3.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$9 {(3 x^{2} + 1)}^{- 4} (6 x + 0)$

Step 1.3.7

Simplify the expression.

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Step 1.3.7.1

Add $6 x$ and $0$ .

$9 {(3 x^{2} + 1)}^{- 4} (6 x)$

Step 1.3.7.2

Multiply $6$ by $9$ .

$54 {(3 x^{2} + 1)}^{- 4} x$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$54 \frac{1}{{(3 x^{2} + 1)}^{4}} x$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Combine $54$ and $\frac{1}{{(3 x^{2} + 1)}^{4}}$ .

$\frac{54}{{(3 x^{2} + 1)}^{4}} x$

Step 1.4.2.2

Combine $\frac{54}{{(3 x^{2} + 1)}^{4}}$ and $x$ .

$\frac{54 x}{{(3 x^{2} + 1)}^{4}}$

Step 1.5

Evaluate the derivative at $x = 0$ .

$\frac{54 (0)}{{(3 {(0)}^{2} + 1)}^{4}}$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $54$ by $0$ .

$\frac{0}{{(3 \cdot 0^{2} + 1)}^{4}}$

Step 1.6.2

Simplify the denominator.

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Step 1.6.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{{(3 \cdot 0 + 1)}^{4}}$

Step 1.6.2.2

Multiply $3$ by $0$ .

$\frac{0}{{(0 + 1)}^{4}}$

Step 1.6.2.3

Add $0$ and $1$ .

$\frac{0}{1^{4}}$

Step 1.6.2.4

One to any power is one.

$\frac{0}{1}$

Step 1.6.3

Divide $0$ by $1$ .

$0$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $0$ and a given point $(0, - 3)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = 0 \cdot (x - (0))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y + 3 = 0 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $0 \cdot (x + 0)$ .

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Step 2.3.1.1

Add $x$ and $0$ .

$y + 3 = 0 \cdot x$

Step 2.3.1.2

Multiply $0$ by $x$ .

$y + 3 = 0$

Step 2.3.2

Subtract $3$ from both sides of the equation.

$y = - 3$

Step 3