Calculus Examples

2 x^{2} + 3 x y + 4 y^{2} - 5 x + 2 y

$2 x^{2} + 3 x y + 4 y^{2} - 5 x + 2 y$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at

$x = x + h$ .

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Step 2.1.1

Replace the variable

$x$ with

$x + h$ in the expression.

$f (x + h) = 2 {(x + h)}^{2} + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite

${(x + h)}^{2}$ as

$(x + h) (x + h)$ .

$f (x + h) = 2 ((x + h) (x + h)) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.2

Expand

$(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 2 (x (x + h) + h (x + h)) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h (x + h)) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h x + h \cdot h) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply

$x$ by

$x$ .

$f (x + h) = 2 (x^{2} + x h + h x + h \cdot h) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.3.1.2

Multiply

$h$ by

$h$ .

$f (x + h) = 2 (x^{2} + x h + h x + h^{2}) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.3.2

Add

$x h$ and

$h x$ .

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Step 2.1.2.1.3.2.1

Reorder

$x$ and

$h$ .

$f (x + h) = 2 (x^{2} + h x + h x + h^{2}) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.3.2.2

Add

$h x$ and

$h x$ .

$f (x + h) = 2 (x^{2} + 2 h x + h^{2}) + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = 2 x^{2} + 2 (2 h x) + 2 h^{2} + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.5

Multiply

$2$ by

$2$ .

$f (x + h) = 2 x^{2} + 4 (h x) + 2 h^{2} + 3 (x + h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.6

Apply the distributive property.

$f (x + h) = 2 x^{2} + 4 h x + 2 h^{2} + (3 x + 3 h) \cdot y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.7

Apply the distributive property.

$f (x + h) = 2 x^{2} + 4 h x + 2 h^{2} + 3 x y + 3 h y + 4 y^{2} - 5 (x + h) + 2 y$

Step 2.1.2.1.8

Apply the distributive property.

$f (x + h) = 2 x^{2} + 4 h x + 2 h^{2} + 3 x y + 3 h y + 4 y^{2} - 5 x - 5 h + 2 y$

Step 2.1.2.2

The final answer is

$2 x^{2} + 4 h x + 2 h^{2} + 3 x y + 3 h y + 4 y^{2} - 5 x - 5 h + 2 y$ .

$2 x^{2} + 4 h x + 2 h^{2} + 3 x y + 3 h y + 4 y^{2} - 5 x - 5 h + 2 y$

Step 2.2

Reorder.

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Step 2.2.1

Move

$- 5 x$ .

$2 x^{2} + 4 h x + 2 h^{2} + 3 x y + 3 h y + 4 y^{2} - 5 h - 5 x + 2 y$

Step 2.2.2

Move

$3 x y$ .

$2 x^{2} + 4 h x + 2 h^{2} + 3 h y + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y$

Step 2.2.3

Move

$2 x^{2}$ .

$4 h x + 2 h^{2} + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y$

Step 2.2.4

Reorder

$4 h x$ and

$2 h^{2}$ .

$2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y$

Step 2.3

Find the components of the definition.

$f (x + h) = 2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y$

$f (x) = 2 x^{2} + 3 x y + 4 y^{2} - 5 x + 2 y$

$f (x + h) = 2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y$

$f (x) = 2 x^{2} + 3 x y + 4 y^{2} - 5 x + 2 y$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - (2 x^{2} + 3 x y + 4 y^{2} - 5 x + 2 y)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - (2 x^{2}) - (3 x y) - (4 y^{2}) - (- 5 x) - (2 y)}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply

$2$ by

$- 1$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - 2 x^{2} - (3 x y) - (4 y^{2}) - (- 5 x) - (2 y)}{h}$

Step 4.1.2.2

Multiply

$3$ by

$- 1$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - 2 x^{2} - 3 (x y) - (4 y^{2}) - (- 5 x) - (2 y)}{h}$

Step 4.1.2.3

Multiply

$4$ by

$- 1$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - 2 x^{2} - 3 (x y) - 4 y^{2} - (- 5 x) - (2 y)}{h}$

Step 4.1.2.4

Multiply

$- 5$ by

$- 1$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - 2 x^{2} - 3 (x y) - 4 y^{2} + 5 x - (2 y)}{h}$

Step 4.1.2.5

Multiply

$2$ by

$- 1$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 2 x^{2} + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - 2 x^{2} - 3 (x y) - 4 y^{2} + 5 x - 2 y}{h}$

Step 4.1.3

Subtract

$2 x^{2}$ from

$2 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y + 0 - 3 x y - 4 y^{2} + 5 x - 2 y}{h}$

Step 4.1.4

Add

$2 h^{2}$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 3 x y + 4 y^{2} - 5 h - 5 x + 2 y - 3 x y - 4 y^{2} + 5 x - 2 y}{h}$

Step 4.1.5

Subtract

$3 x y$ from

$3 x y$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 4 y^{2} - 5 h - 5 x + 2 y + 0 - 4 y^{2} + 5 x - 2 y}{h}$

Step 4.1.6

Add

$2 h^{2}$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y + 4 y^{2} - 5 h - 5 x + 2 y - 4 y^{2} + 5 x - 2 y}{h}$

Step 4.1.7

Subtract

$4 y^{2}$ from

$4 y^{2}$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y - 5 h - 5 x + 2 y + 0 + 5 x - 2 y}{h}$

Step 4.1.8

Add

$2 h^{2}$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y - 5 h - 5 x + 2 y + 5 x - 2 y}{h}$

Step 4.1.9

Add

$- 5 x$ and

$5 x$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y - 5 h + 2 y + 0 - 2 y}{h}$

Step 4.1.10

Add

$2 h^{2}$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y - 5 h + 2 y - 2 y}{h}$

Step 4.1.11

Subtract

$2 y$ from

$2 y$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y - 5 h + 0}{h}$

Step 4.1.12

Add

$2 h^{2} + 4 h x + 3 h y - 5 h$ and

$0$ .

$f' (x) = lim_{h \to 0} \frac{2 h^{2} + 4 h x + 3 h y - 5 h}{h}$

Step 4.1.13

Factor

$h$ out of

$2 h^{2} + 4 h x + 3 h y - 5 h$ .

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Step 4.1.13.1

Factor

$h$ out of

$2 h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h) + 4 h x + 3 h y - 5 h}{h}$

Step 4.1.13.2

Factor

$h$ out of

$4 h x$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h) + h (4 x) + 3 h y - 5 h}{h}$

Step 4.1.13.3

Factor

$h$ out of

$3 h y$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h) + h (4 x) + h (3 y) - 5 h}{h}$

Step 4.1.13.4

Factor

$h$ out of

$- 5 h$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h) + h (4 x) + h (3 y) + h \cdot - 5}{h}$

Step 4.1.13.5

Factor

$h$ out of

$h (2 h) + h (4 x)$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h + 4 x) + h (3 y) + h \cdot - 5}{h}$

Step 4.1.13.6

Factor

$h$ out of

$h (2 h + 4 x) + h (3 y)$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h + 4 x + 3 y) + h \cdot - 5}{h}$

Step 4.1.13.7

Factor

$h$ out of

$h (2 h + 4 x + 3 y) + h \cdot - 5$ .

$f' (x) = lim_{h \to 0} \frac{h (2 h + 4 x + 3 y - 5)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of

$h$ .

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Step 4.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (2 h + 4 x + 3 y - 5)}{h}$

Step 4.2.1.2

Divide

$2 h + 4 x + 3 y - 5$ by

$1$ .

$f' (x) = lim_{h \to 0} 2 h + 4 x + 3 y - 5$

Step 4.2.2

Move

$2 h$ .

$f' (x) = lim_{h \to 0} 4 x + 3 y + 2 h - 5$

Step 5

Evaluate the limit.

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Step 5.1

Split the limit using the Sum of Limits Rule on the limit as

$h$ approaches

$0$ .

$lim_{h \to 0} 4 x + lim_{h \to 0} 3 y + lim_{h \to 0} 2 h - lim_{h \to 0} 5$

Step 5.2

Evaluate the limit of

$4 x$ which is constant as

$h$ approaches

$0$ .

$4 x + lim_{h \to 0} 3 y + lim_{h \to 0} 2 h - lim_{h \to 0} 5$

Step 5.3

Evaluate the limit of

$3 y$ which is constant as

$h$ approaches

$0$ .

$4 x + 3 y + lim_{h \to 0} 2 h - lim_{h \to 0} 5$

Step 5.4

Move the term

$2$ outside of the limit because it is constant with respect to

$h$ .

$4 x + 3 y + 2 lim_{h \to 0} h - lim_{h \to 0} 5$

Step 5.5

Evaluate the limit of

$5$ which is constant as

$h$ approaches

$0$ .

$4 x + 3 y + 2 lim_{h \to 0} h - 1 \cdot 5$

Step 6

Evaluate the limit of

$h$ by plugging in

$0$ for

$h$ .

$4 x + 3 y + 2 \cdot 0 - 1 \cdot 5$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply

$2$ by

$0$ .

$4 x + 3 y + 0 - 1 \cdot 5$

Step 7.1.2

Multiply

$- 1$ by

$5$ .

$4 x + 3 y + 0 - 5$

Step 7.2

Add

$4 x + 3 y$ and

$0$ .

$4 x + 3 y - 5$

Step 8