Calculus Examples

$x y^{2} + y e^{x^{2}} + 5$

Step 1

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

$\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [y e^{x^{2}}] + \frac{d}{d x} [5]$

Step 2

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

Tap for more steps...

Step 2.1

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

$y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [y e^{x^{2}}] + \frac{d}{d x} [5]$

Step 2.2

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

$y^{2} \cdot 1 + \frac{d}{d x} [y e^{x^{2}}] + \frac{d}{d x} [5]$

Step 2.3

Multiply $y^{2}$ by $1$ .

$y^{2} + \frac{d}{d x} [y e^{x^{2}}] + \frac{d}{d x} [5]$

Step 3

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

Tap for more steps...

Step 3.1

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

$y^{2} + y \frac{d}{d x} [e^{x^{2}}] + \frac{d}{d x} [5]$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$y^{2} + y (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [5]$

Step 3.2.2

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

$y^{2} + y (e^{u} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [5]$

Step 3.2.3

Replace all occurrences of $u$ with $x^{2}$ .

$y^{2} + y (e^{x^{2}} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [5]$

Step 3.3

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

$y^{2} + y e^{x^{2}} (2 x) + \frac{d}{d x} [5]$

Step 4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$y^{2} + y e^{x^{2}} (2 x) + 0$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Add $y^{2} + y e^{x^{2}} (2 x)$ and $0$ .

$y^{2} + y e^{x^{2}} (2 x)$

Step 5.2

Reorder terms.

$y^{2} + 2 e^{x^{2}} x y$

Step 5.3

Reorder factors in $y^{2} + 2 e^{x^{2}} x y$ .

$y^{2} + 2 x y e^{x^{2}}$