Find dy/dx y=3x^3-2x^2+5x-1

$y = 3 x^{3} - 2 x^{2} + 5 x - 1$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

By the Sum Rule, the derivative of $3 x^{3} - 2 x^{2} + 5 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$ .

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

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

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

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

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

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

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$

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

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

$9 x^{2} - 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$

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

$9 x^{2} - 2 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$

Multiply $2$ by $- 2$ .

$9 x^{2} - 4 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$

Evaluate $\frac{d}{d x} [5 x]$ .

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

$9 x^{2} - 4 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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

$9 x^{2} - 4 x + 5 \cdot 1 + \frac{d}{d x} [- 1]$

Multiply $5$ by $1$ .

$9 x^{2} - 4 x + 5 + \frac{d}{d x} [- 1]$

Differentiate using the Constant Rule.

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

$9 x^{2} - 4 x + 5 + 0$

Add $9 x^{2} - 4 x + 5$ and $0$ .

$9 x^{2} - 4 x + 5$

Step 4

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

$y' = 9 x^{2} - 4 x + 5$

Step 5

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

$\frac{d y}{d x} = 9 x^{2} - 4 x + 5$

Find dy/dx y=3x^3-2x^2+5x-1

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