Calculus Examples

$f (x) = x \ln (x) - x$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

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.

$f (x + h) = (x + h) \ln (x + h) - (x + h)$

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

Apply the distributive property.

$f (x + h) = x \ln (x + h) + h \ln (x + h) - (x + h)$

Step 2.1.2.1.2

Apply the distributive property.

$f (x + h) = x \ln (x + h) + h \ln (x + h) - x - h$

Step 2.1.2.2

The final answer is $x \ln (x + h) + h \ln (x + h) - x - h$ .

$x \ln (x + h) + h \ln (x + h) - x - h$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $x$ and $h$ .

$x \ln (h + x) + h \ln (x + h) - x - h$

Step 2.2.2

Reorder $x$ and $h$ .

$x \ln (h + x) + h \ln (h + x) - x - h$

Step 2.2.3

Move $h \ln (h + x)$ .

$x \ln (h + x) - x - h + h \ln (h + x)$

Step 2.2.4

Move $x \ln (h + x)$ .

$- x - h + x \ln (h + x) + h \ln (h + x)$

Step 2.2.5

Reorder $- x$ and $- h$ .

$- h - x + x \ln (h + x) + h \ln (h + x)$

Step 2.3

Find the components of the definition.

$f (x + h) = - h - x + x \ln (h + x) + h \ln (h + x)$

$f (x) = (x) \ln (x) - x$

$f (x + h) = - h - x + x \ln (h + x) + h \ln (h + x)$

$f (x) = (x) \ln (x) - x$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- h - x + x \ln (h + x) + h \ln (h + x) - ((x) \ln (x) - x)}{h}$

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.

$\frac{- h - x + x \ln (h + x) + h \ln (h + x) - ((x) \ln (x)) - - x}{h}$

Step 4.1.2

Multiply $- - x$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- h - x + x \ln (h + x) + h \ln (h + x) - x \ln (x) + 1 x}{h}$

Step 4.1.2.2

Multiply $x$ by $1$ .

$\frac{- h - x + x \ln (h + x) + h \ln (h + x) - x \ln (x) + x}{h}$

Step 4.1.3

Add $- x$ and $x$ .

$\frac{- h + x \ln (h + x) + h \ln (h + x) - x \ln (x) + 0}{h}$

Step 4.1.4

Add $- h + x \ln (h + x) + h \ln (h + x) - x \ln (x)$ and $0$ .

$\frac{- h + x \ln (h + x) + h \ln (h + x) - x \ln (x)}{h}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Factor $- 1$ out of $x \ln (h + x)$ .

$\frac{- h - (- x \ln (h + x)) + h \ln (h + x) - x \ln (x)}{h}$

Step 4.2.2

Factor $- 1$ out of $- h - (- x \ln (h + x))$ .

$\frac{- (h - x \ln (h + x)) + h \ln (h + x) - x \ln (x)}{h}$

Step 4.2.3

Factor $- 1$ out of $h \ln (h + x)$ .

$\frac{- (h - x \ln (h + x)) - (- h \ln (h + x)) - x \ln (x)}{h}$

Step 4.2.4

Factor $- 1$ out of $- (h - x \ln (h + x)) - (- h \ln (h + x))$ .

$\frac{- (h - x \ln (h + x) - h \ln (h + x)) - x \ln (x)}{h}$

Step 4.2.5

Factor $- 1$ out of $- x \ln (x)$ .

$\frac{- (h - x \ln (h + x) - h \ln (h + x)) - (x \ln (x))}{h}$

Step 4.2.6

Factor $- 1$ out of $- (h - x \ln (h + x) - h \ln (h + x)) - (x \ln (x))$ .

$\frac{- (h - x \ln (h + x) - h \ln (h + x) + x \ln (x))}{h}$

Step 4.2.7

Simplify the expression.

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Step 4.2.7.1

Rewrite $- (h - x \ln (h + x) - h \ln (h + x) + x \ln (x))$ as $- 1 (h - x \ln (h + x) - h \ln (h + x) + x \ln (x))$ .

$\frac{- 1 (h - x \ln (h + x) - h \ln (h + x) + x \ln (x))}{h}$

Step 4.2.7.2

Move the negative in front of the fraction.

$- \frac{h - x \ln (h + x) - h \ln (h + x) + x \ln (x)}{h}$

Step 5