Calculus Examples

$y = 3^{x} x^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3^{x} x^{3})$

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 Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 3^{x}$ and $g (x) = x^{3}$ .

$3^{x} \frac{d}{d x} [x^{3}] + x^{3} \frac{d}{d x} [3^{x}]$

Step 3.2

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

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

Step 3.3

Multiply $3^{x}$ by $3$ by adding the exponents.

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Step 3.3.1

Move $3$ .

$3 \cdot 3^{x} x^{2} + x^{3} \frac{d}{d x} [3^{x}]$

Step 3.3.2

Multiply $3$ by $3^{x}$ .

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Step 3.3.2.1

Raise $3$ to the power of $1$ .

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

Step 3.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$3^{1 + x} x^{2} + x^{3} (3^{x} \ln (3))$

Step 3.5

Simplify.

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Step 3.5.1

Reorder terms.

$3^{1 + x} x^{2} + 3^{x} x^{3} \ln (3)$

Step 3.5.2

Reorder factors in $3^{1 + x} x^{2} + 3^{x} x^{3} \ln (3)$ .

$x^{2} \cdot 3^{1 + x} + x^{3} \cdot 3^{x} \ln (3)$

Step 4

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

$y' = x^{2} \cdot 3^{1 + x} + x^{3} \cdot (3^{x} \ln (3))$

Step 5

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

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