Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $f (x) = x^{3}$ and $g (x) = {(2 x - 1)}^{5}$ .

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

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 $f (x) = x^{5}$ and $g (x) = 2 x - 1$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $2 x - 1$ .

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$x^{3} (5 u^{4} \frac{d}{d x} [2 x - 1]) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.2.3

Replace all occurrences of $u$ with $2 x - 1$ .

$x^{3} (5 {(2 x - 1)}^{4} \frac{d}{d x} [2 x - 1]) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3

Differentiate.

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Step 3.3.1

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

$x^{3} (5 {(2 x - 1)}^{4} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1])) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.2

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

$x^{3} (5 {(2 x - 1)}^{4} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.3

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

$x^{3} (5 {(2 x - 1)}^{4} (2 \cdot 1 + \frac{d}{d x} [- 1])) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.4

Multiply $2$ by $1$ .

$x^{3} (5 {(2 x - 1)}^{4} (2 + \frac{d}{d x} [- 1])) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.5

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

$x^{3} (5 {(2 x - 1)}^{4} (2 + 0)) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $2$ and $0$ .

$x^{3} (5 {(2 x - 1)}^{4} \cdot 2) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.6.2

Multiply $2$ by $5$ .

$x^{3} (10 {(2 x - 1)}^{4}) + {(2 x - 1)}^{5} \frac{d}{d x} [x^{3}]$

Step 3.3.7

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

$x^{3} (10 {(2 x - 1)}^{4}) + {(2 x - 1)}^{5} (3 x^{2})$

Step 3.3.8

Move $3$ to the left of ${(2 x - 1)}^{5}$ .

$x^{3} (10 {(2 x - 1)}^{4}) + 3 \cdot {(2 x - 1)}^{5} x^{2}$

Step 3.4

Simplify.

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Step 3.4.1

Factor $x^{2} {(2 x - 1)}^{4}$ out of $x^{3} (10 {(2 x - 1)}^{4}) + 3 {(2 x - 1)}^{5} x^{2}$ .

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Step 3.4.1.1

Factor $x^{2} {(2 x - 1)}^{4}$ out of $x^{3} (10 {(2 x - 1)}^{4})$ .

$x^{2} {(2 x - 1)}^{4} (x \cdot (10)) + 3 {(2 x - 1)}^{5} x^{2}$

Step 3.4.1.2

Factor $x^{2} {(2 x - 1)}^{4}$ out of $3 {(2 x - 1)}^{5} x^{2}$ .

$x^{2} {(2 x - 1)}^{4} (x \cdot (10)) + x^{2} {(2 x - 1)}^{4} (3 (2 x - 1))$

Step 3.4.1.3

Factor $x^{2} {(2 x - 1)}^{4}$ out of $x^{2} {(2 x - 1)}^{4} (x \cdot (10)) + x^{2} {(2 x - 1)}^{4} (3 (2 x - 1))$ .

$x^{2} {(2 x - 1)}^{4} (x \cdot (10) + 3 (2 x - 1))$

Step 3.4.2

Move $10$ to the left of $x$ .

$x^{2} {(2 x - 1)}^{4} (10 x + 3 (2 x - 1))$

Step 4

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

$y' = x^{2} {(2 x - 1)}^{4} (10 x + 3 (2 x - 1))$

Step 5

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

$\frac{d y}{d x} = x^{2} {(2 x - 1)}^{4} (10 x + 3 (2 x - 1))$