UNIT 6 • CALCULUS

Derivative Calculator
Power, Product, Quotient & Chain Rules

A derivative measures how fast a function changes. You find it rule by rule: $\frac{d}{dx}x^n = nx^{n-1}$ for powers, then the product, quotient and chain rules to combine pieces. This calculator shows every rule it uses.

THE CORE RULESd/dx

\begin{gathered} (x^n)' = n x^{n-1} \qquad (uv)' = u'v + uv' \\ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \qquad \frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x) \end{gathered}

1. Spot the Outer ShapeIs the whole thing a sum, a product, a quotient, or a function of something?

2. Apply That RuleWrite it out with u and v, leaving their derivatives to work out next.

3. Go InsideDifferentiate each piece the same way, then tidy up.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Differentiate $y = x^2 \sin(3x)$ .

STEP 01 • IT IS A PRODUCT

Take $u = x^2$ and $v = \sin(3x)$ , and use $(uv)' = u'v + uv'$ .

STEP 02 • DIFFERENTIATE EACH PART

u' = \frac{d}{dx}x^2 = 2x \quad \text{(power rule)}

v' = \frac{d}{dx}\sin(3x) = \cos(3x) \cdot 3 = 3\cos(3x) \quad \text{(chain rule)}

STEP 03 • PUT IT TOGETHER

\frac{dy}{dx} = 2x \cdot \sin(3x) + x^2 \cdot 3\cos(3x) = 2x\sin(3x) + 3x^2\cos(3x)

FINAL CONCLUSION

dy/dx = 2x sin(3x) + 3x² cos(3x)

\frac{dy}{dx} = 2x\sin(3x) + 3x^2\cos(3x)

The calculator below opens on this function and shows the same rules, then checks the answer numerically.

LABORATORY

Interactive Derivative Calculator

LABORATORY

Derivative Engine

f(x) =

Try:

Derivative

Slope at x = (optional)

Use ^ for powers and brackets for functions: sin(3x), ln(x), e^(2x), sqrt(x).

Common Questions

What is a derivative?

The rate at which a function changes: the slope of its graph at each point.

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

. For

f(x) = x^2

f'(x) = 2x

, so the slope at x = 3 is 6.

When do you use the chain rule?

Whenever one function sits inside another, like

\sin(x^3)

e^{2x}

. Differentiate the outside, keep the inside as it is, then multiply by the derivative of the inside:

\frac{d}{dx}\sin(x^3) = \cos(x^3) \cdot 3x^2

What is the difference between the product rule and the quotient rule?

Product:

(uv)' = u'v + uv'

. Quotient:

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

. The order matters in the quotient rule because of the minus sign.

What are the derivatives of eˣ and ln x?

\frac{d}{dx}e^x = e^x

(it is its own derivative) and

\frac{d}{dx}\ln x = \frac{1}{x}

for x > 0. With the chain rule:

\frac{d}{dx}e^{u} = e^{u}u'

and

\frac{d}{dx}\ln u = \frac{u'}{u}

PreviousLimit Calculator

Unit 6•46 of 63

NextQuadratic Equation

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 6 • CALCULUS

Derivative Calculator
Power, Product, Quotient & Chain Rules

THE CORE RULESd/dx

\begin{gathered} (x^n)' = n x^{n-1} \qquad (uv)' = u'v + uv' \\ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \qquad \frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x) \end{gathered}

1. Spot the Outer ShapeIs the whole thing a sum, a product, a quotient, or a function of something?

2. Apply That RuleWrite it out with u and v, leaving their derivatives to work out next.

3. Go InsideDifferentiate each piece the same way, then tidy up.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Differentiate $y = x^2 \sin(3x)$ .

STEP 01 • IT IS A PRODUCT

Take $u = x^2$ and $v = \sin(3x)$ , and use $(uv)' = u'v + uv'$ .

STEP 02 • DIFFERENTIATE EACH PART

u' = \frac{d}{dx}x^2 = 2x \quad \text{(power rule)}

v' = \frac{d}{dx}\sin(3x) = \cos(3x) \cdot 3 = 3\cos(3x) \quad \text{(chain rule)}

STEP 03 • PUT IT TOGETHER

\frac{dy}{dx} = 2x \cdot \sin(3x) + x^2 \cdot 3\cos(3x) = 2x\sin(3x) + 3x^2\cos(3x)

FINAL CONCLUSION

dy/dx = 2x sin(3x) + 3x² cos(3x)

\frac{dy}{dx} = 2x\sin(3x) + 3x^2\cos(3x)

The calculator below opens on this function and shows the same rules, then checks the answer numerically.

LABORATORY

Interactive Derivative Calculator

LABORATORY

Derivative Engine

f(x) =

Try:

Derivative

Slope at x = (optional)

Use ^ for powers and brackets for functions: sin(3x), ln(x), e^(2x), sqrt(x).

Common Questions

What is a derivative?

The rate at which a function changes: the slope of its graph at each point.

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

. For

f(x) = x^2

f'(x) = 2x

, so the slope at x = 3 is 6.

When do you use the chain rule?

Whenever one function sits inside another, like

\sin(x^3)

e^{2x}

. Differentiate the outside, keep the inside as it is, then multiply by the derivative of the inside:

\frac{d}{dx}\sin(x^3) = \cos(x^3) \cdot 3x^2

What is the difference between the product rule and the quotient rule?

Product:

(uv)' = u'v + uv'

. Quotient:

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

. The order matters in the quotient rule because of the minus sign.

What are the derivatives of eˣ and ln x?

\frac{d}{dx}e^x = e^x

(it is its own derivative) and

\frac{d}{dx}\ln x = \frac{1}{x}

for x > 0. With the chain rule:

\frac{d}{dx}e^{u} = e^{u}u'

and

\frac{d}{dx}\ln u = \frac{u'}{u}

PreviousLimit Calculator

Unit 6•46 of 63

NextQuadratic Equation

Derivative Calculator Power, Product, Quotient & Chain Rules

Step-by-Step Worked Example

dy/dx = 2x sin(3x) + 3x² cos(3x)

Interactive Derivative Calculator

Derivative Engine

Common Questions

NETZ Engine

Derivative Calculator Power, Product, Quotient & Chain Rules

Step-by-Step Worked Example

dy/dx = 2x sin(3x) + 3x² cos(3x)

Interactive Derivative Calculator

Derivative Engine

Common Questions

Derivative Calculator
Power, Product, Quotient & Chain Rules

Derivative Calculator
Power, Product, Quotient & Chain Rules