Calculus Examples

$y = p x^{2}$

Step 1

Consider the limit definition of the derivative.

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = p + h$ .

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Step 2.1.1

Replace the variable $p$ with $p + h$ in the expression.

$f (p + h) = (p + h) \cdot x^{2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (p + h) = p x^{2} + h x^{2}$

Step 2.1.2.2

The final answer is $p x^{2} + h x^{2}$ .

$p x^{2} + h x^{2}$

Step 2.2

Find the components of the definition.

$f (p + h) = p x^{2} + h x^{2}$

$f (p) = p x^{2}$

$f (p + h) = p x^{2} + h x^{2}$

$f (p) = p x^{2}$

Step 3

Plug in the components.

$f' (p) = lim_{h \to 0} \frac{p x^{2} + h x^{2} - (p x^{2})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Subtract $p x^{2}$ from $p x^{2}$ .

$f' (p) = lim_{h \to 0} \frac{h x^{2} + 0}{h}$

Step 4.1.2

Add $h x^{2}$ and $0$ .

$f' (p) = lim_{h \to 0} \frac{h x^{2}}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$f' (p) = lim_{h \to 0} \frac{h x^{2}}{h}$

Step 4.2.2

Divide $x^{2}$ by $1$ .

$f' (p) = lim_{h \to 0} x^{2}$

Step 5

Evaluate the limit of $x^{2}$ which is constant as $h$ approaches $0$ .

$x^{2}$

Step 6