Unit 2 · Lesson 2 of 11

Use derivative rules to measure sensitivity.

A nonlinear response can become much more sensitive as its input grows. Compare its value with its local slope.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its derivative value.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase input x? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

f(x)=x^n,\quad f'(x)=nx^{n-1}

Positive inputs and integer powers. The derivative curve is shown alongside the function.

FunctionDerivative · dashed

Input x2 Power n3

derivative value

f'(2)=3(2)^{3-1}=12

The power rule predicts how sensitive the output is to a small input change. Slope and output are different quantities.

Assumptions, units, and limits

Positive inputs and integer powers. The derivative curve is shown alongside the function.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

The power rule follows from the difference quotient for integer powers. Constant multiples and sums differentiate term by term. A constant shift in output does not affect the slope; a horizontal rescaling does.

f(x)=x^n,\quad f'(x)=nx^{n-1}

Keep the reasoning visible

\frac{d}{dx}(af+bg+c)=af'+bg'

The power rule predicts how sensitive the output is to a small input change. Slope and output are different quantities.

What makes this model useful?

Positive inputs and integer powers. The derivative curve is shown alongside the function.

A derivative is not the function divided by its input. Preserve the variable and its units while applying rules.

A worked example

Calculate the starting scenario

Use input x = 2 ; power n = 3 .
$f'(2)=3(2)^{3-1}=12$
The power rule predicts how sensitive the output is to a small input change. Slope and output are different quantities. The checked quantity is derivative value.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the derivative value using input x 2 , power n 3 . State the meaning of its sign and units.

Checked quantity: derivative value · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

f(x)=x^n,\quad f'(x)=nx^{n-1}

Positive inputs and integer powers. The derivative curve is shown alongside the function.

FunctionDerivative · dashed

Input x2 Power n3

derivative value

f'(2)=3(2)^{3-1}=12

The power rule predicts how sensitive the output is to a small input change. Slope and output are different quantities.

Assumptions, units, and limits

Positive inputs and integer powers. The derivative curve is shown alongside the function.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

f(x)=x^n,\quad f'(x)=nx^{n-1}

Calculate and explain

Apply the idea to a new scenario

For $f=x$ ⁴ at $x=1.5,$ find the derivative.

Checked quantity: derivative value · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

f(x)=x^n,\quad f'(x)=nx^{n-1}

Positive inputs and integer powers. The derivative curve is shown alongside the function.

FunctionDerivative · dashed

Input x1.5 Power n4

derivative value

f'(1.5)=4(1.5)^{4-1}=13.5

The power rule predicts how sensitive the output is to a small input change. Slope and output are different quantities.

Assumptions, units, and limits

Positive inputs and integer powers. The derivative curve is shown alongside the function.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

f(x)=x^n,\quad f'(x)=nx^{n-1}

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Read the model

f(x)=x^n,\quad f'(x)=nx^{n-1}

Positive inputs and integer powers. The derivative curve is shown alongside the function.

FunctionDerivative · dashed

Input x2 Power n3

derivative value

f'(2)=3(2)^{3-1}=12

The power rule predicts how sensitive the output is to a small input change. Slope and output are different quantities.

Assumptions, units, and limits

Positive inputs and integer powers. The derivative curve is shown alongside the function.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. The power rule follows from the difference quotient for integer powers. Constant multiples and sums differentiate term by term. A constant shift in output does not affect the slope; a horizontal rescaling does.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.