Calculus Examples

$y = 4 x^{3} - 7 x^{2} + 3 x - 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (4 x^{3} - 7 x^{2} + 3 x - 2)$

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

By the Sum Rule, the derivative of $4 x^{3} - 7 x^{2} + 3 x - 2$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 7 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.3

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Step 3.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - 7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.3.2

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

$12 x^{2} - 7 (2 x) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.3.3

Multiply $2$ by $- 7$ .

$12 x^{2} - 14 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$

Step 3.4

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 3.4.1

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

$12 x^{2} - 14 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 3.4.2

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

$12 x^{2} - 14 x + 3 \cdot 1 + \frac{d}{d x} [- 2]$

Step 3.4.3

Multiply $3$ by $1$ .

$12 x^{2} - 14 x + 3 + \frac{d}{d x} [- 2]$

Step 3.5

Differentiate using the Constant Rule.

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Step 3.5.1

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

$12 x^{2} - 14 x + 3 + 0$

Step 3.5.2

Add $12 x^{2} - 14 x + 3$ and $0$ .

$12 x^{2} - 14 x + 3$

Step 4

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

$y' = 12 x^{2} - 14 x + 3$

Step 5

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

$\frac{d y}{d x} = 12 x^{2} - 14 x + 3$