Calculus Examples

$x^{3} + {(t x)}^{3} + 3 x (t x)$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $x^{3} + {(t x)}^{3} + 3 x (t x)$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [{(t x)}^{3}] + \frac{d}{d x} [3 x (t x)]$ .

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = t x$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $t x$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $t x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $t$ by $1$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 3.5

Since $3 t$ is constant with respect to $x$ , the derivative of $3 t x^{2}$ with respect to $x$ is $3 t \frac{d}{d x} [x^{2}]$ .

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

Step 3.6

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

$3 x^{2} + 3 {(t x)}^{2} t + 3 t (2 x)$

Step 3.7

Multiply $2$ by $3$ .

$3 x^{2} + 3 {(t x)}^{2} t + 6 t x$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the product rule to $t x$ .

$3 x^{2} + 3 (t^{2} x^{2}) t + 6 t x$

Step 4.2

Combine terms.

Tap for more steps...

Step 4.2.1

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

$3 x^{2} + 3 (t^{1} t^{2}) x^{2} + 6 t x$

Step 4.2.2

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

$3 x^{2} + 3 t^{1 + 2} x^{2} + 6 t x$

Step 4.2.3

Add $1$ and $2$ .

$3 x^{2} + 3 t^{3} x^{2} + 6 t x$