Product Rule

Step 1: Simplify the expression

This expression is already simplified.

2x(e^x)

Step 2: Apply the product rule.

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$

$\frac{d}{dx}\left[2x\left(e^x\right)\right]$

$2x\frac{d}{dx}{e}^x+e^x\frac{d}{dx}2x$

Step 3: Take the derivative of each part.

Use the natural exponential rule (NER) to differentiate e^x

Use the constant multiple and power rules (CM/P) to differentiate 2x.

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$\frac{d}{dx}2x=2\frac{d}{dx}x=2$    CM/P

Step 4: Substitute the derivatives into the product rule & simplify.

$2x\left(e^x\right)+e^x\left(2\right)$

$2e^x\left(x+1\right)$

Step 1: Apply the product rule.

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$

$\frac{d}{dx}\left[x^2\left(6+9x\right)\right]$

$x^2\frac{d}{dx}\left[\left(6+9x\right)\right]+\left(6+9x\right)\frac{d}{dx}{x}^2$

Step 2: Take the derivative of each part.

To differentiate 6 + 9x apply the sum rule, the constant multiple rule and then the constant and power rules.

To differentiate x²apply the power rule.

$\frac{d}{dx}6+9x$

$\frac{d}{dx}6+\frac{d}{dx}9x$ Sum rule

$\frac{d}{dx}6+9\frac{d}{dx}x$ CM

$0+9=9$     C&P

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$\frac{d}{dx}{x}^2=2x$

Step 3: Substitute the derivatives & simplify.

$x^2\left(9\right)+\left(6+9x\right)\left(2x\right)$

$9x^2+12x+18x^2$

$27x^2+12x$

Step 1: Simplify first.

6x² + 9x³

Step 2: Apply the sum rule.

$\frac{d}{dx}\left[6x^2+9x^3\right]$

$\frac{d}{dx}6x^2+\frac{d}{dx}9x^3$

Step 3: Take the derivative of each part.

To differentiate 6x² apply the constant multiple and power rules.

To differentiate 9x³ apply the constant multiple and power rules.

$\frac{d}{dx}6x^2$

$6\frac{d}{dx}{x}^2$          CM

$6\left(2x\right)=12x$ Power

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$\frac{d}{dx}9x^3$

$9\frac{d}{dx}{x}^3$        CM

$9\left(3x^2\right)=27x^2$ Power

Step 4: Add/Subtract the derivatives & simplify.

Write polynomials in descending order.

$12x+27x^2$

$27x^2+12x$

Step 1: Apply the product rule.

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$

$\frac{d}{dx}\left[\left(x^5+6\right)\left(\frac{1}{2}{x}^4-1\right)\right]$

$\left(x^5+6\right)\frac{d}{dx}\left[\frac{1}{2}{x}^4-1\right]+\left(\frac{1}{2}{x}^4-1\right)\frac{d}{dx}\left(x^5+6\right)$

Step 2: Take the derivative of each part.

To differentiate $\left(\frac{1}{2}{x}^4-1\right)$ apply the difference rule, the constant multiple rule and then the constant and power rules.

To differentiate $\left(x^5+6\right)$, apply the sum rule and then the constant and power rules.

$\frac{d}{dx}\frac{1}{2}{x}^4-1$     Original

$\frac{1}{2}\frac{d}{dx}{x}^4-\frac{d}{dx}1$ Diff. Rule & CM

$\frac{4}{2}{x}^3-0=2x^3$  Constant & Power

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$\frac{d}{dx}\left(x^5+6\right)$ Original

$\frac{d}{dx}{x}^5+\frac{d}{dx}6$ Sum Rule

$5x^4$ Constant & Power

Step 3: Substitute the derivatives & simplify.

$\left(x^5+6\right)\left(2x^3\right)+\left(\frac{1}{2}{x}^4-1\right)\left(5x^4\right)$

$2x^8+12x^3+\frac{5}{2}{x}^8-5x^4$

$4.5x^8-5x^4+12x^3$

Step 1: Simplify first.

Distribute the x².

$\frac{1}{2}{x}^9-x^5+3x^4-6$

Step 2: Apply the sum/difference rule.

$\frac{d}{dx}\left[\frac{1}{2}{x}^9-x^5+3x^4-6 \right]$

$\frac{d}{dx}\frac{1}{2}{x}^9-\frac{d}{dx}{x}^5+\frac{d}{dx}3x^4-\frac{d}{dx}6$

Step 3: Take the derivative of each part.

To differentiate $\left(\frac{1}{2}{x}^9\right)$ apply the constant multiple rule and then the power rule.

To differentiate (x⁵), apply the power rule.

To differentiate (3x⁴) apply the constant multiple rule and then the power rule.

To differentiate (6) apply the constant rule.

$\frac{d}{dx}\frac{1}{2}{x}^9=\frac{1}{2}\frac{d}{dx}{x}^9=\frac{1}{2}\left(9x^8\right)=\frac{9}{2}{x}^8=4.5x^8$

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$\frac{d}{dx}{x}^5=5x^4$

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$\frac{d}{dx}3x^4=3\frac{d}{dx}{x}^4=3\left(4x^3\right)=12x^3$

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$\frac{d}{dx}6=0$

Step 4: Add/Subtract the derivatives & simplify.

$4.5x^8-5x^4+12x^3$