Calculus Examples

$y = \ln (3 x^{2} - 5 x + 2)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (3 x^{2} - 5 x + 2))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x^{2} - 5 x + 2$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 5 x + 2$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x^{2} - 5 x + 2]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [3 x^{2} - 5 x + 2]$

Step 3.1.3

Replace all occurrences of $u$ with $3 x^{2} - 5 x + 2$ .

$\frac{1}{3 x^{2} - 5 x + 2} \frac{d}{d x} [3 x^{2} - 5 x + 2]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $3 x^{2} - 5 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [2]$ .

$\frac{1}{3 x^{2} - 5 x + 2} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [2])$

Step 3.2.2

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

$\frac{1}{3 x^{2} - 5 x + 2} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [2])$

Step 3.2.3

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

$\frac{1}{3 x^{2} - 5 x + 2} (3 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [2])$

Step 3.2.4

Multiply $2$ by $3$ .

$\frac{1}{3 x^{2} - 5 x + 2} (6 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [2])$

Step 3.2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$\frac{1}{3 x^{2} - 5 x + 2} (6 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 3.2.6

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

$\frac{1}{3 x^{2} - 5 x + 2} (6 x - 5 \cdot 1 + \frac{d}{d x} [2])$

Step 3.2.7

Multiply $- 5$ by $1$ .

$\frac{1}{3 x^{2} - 5 x + 2} (6 x - 5 + \frac{d}{d x} [2])$

Step 3.2.8

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

$\frac{1}{3 x^{2} - 5 x + 2} (6 x - 5 + 0)$

Step 3.2.9

Add $6 x - 5$ and $0$ .

$\frac{1}{3 x^{2} - 5 x + 2} (6 x - 5)$

Step 3.3

Simplify.

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Step 3.3.1

Reorder the factors of $\frac{1}{3 x^{2} - 5 x + 2} (6 x - 5)$ .

$(6 x - 5) \frac{1}{3 x^{2} - 5 x + 2}$

Step 3.3.2

Factor by grouping.

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Step 3.3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = - 5$ .

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Step 3.3.2.1.1

Factor $- 5$ out of $- 5 x$ .

$(6 x - 5) \frac{1}{3 x^{2} - 5 (x) + 2}$

Step 3.3.2.1.2

Rewrite $- 5$ as $- 2$ plus $- 3$

$(6 x - 5) \frac{1}{3 x^{2} + (- 2 - 3) x + 2}$

Step 3.3.2.1.3

Apply the distributive property.

$(6 x - 5) \frac{1}{3 x^{2} - 2 x - 3 x + 2}$

Step 3.3.2.2

Factor out the greatest common factor from each group.

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Step 3.3.2.2.1

Group the first two terms and the last two terms.

$(6 x - 5) \frac{1}{(3 x^{2} - 2 x) - 3 x + 2}$

Step 3.3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$(6 x - 5) \frac{1}{x (3 x - 2) - (3 x - 2)}$

Step 3.3.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(6 x - 5) \frac{1}{(3 x - 2) (x - 1)}$

Step 3.3.3

Multiply $6 x - 5$ by $\frac{1}{(3 x - 2) (x - 1)}$ .

$\frac{6 x - 5}{(3 x - 2) (x - 1)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{6 x - 5}{(3 x - 2) (x - 1)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{6 x - 5}{(3 x - 2) (x - 1)}$