Chain-rule branch and product-rule branch recombined at the derivativeUpdates with every input
Math & Statistics
Derivative Step-by-Step Calculator
Differentiate f(x)=(ax+b)^p(cx+d) by exposing the inner derivative, outer power rule, product-rule branches, and final evaluation instead of collapsing the reasoning into one line.
Inner value u(x0) -
Linear factor v(x0) -
u'(x0) -
v'(x0) -
u'v branch -
uv' branch -
Final f'(x0) -
f(x0) -
RULE DEPENDENCY MAP
Chain-rule branch and product-rule branch recombined at the derivative
A live node tree sizes each branch by its signed contribution so cancellation and dominance remain visible.
RULE-BY-RULE WORKSHEET
Symbolic transformation, substitution, and contribution at each node
The ledger preserves the two product-rule terms until the final addition.
| Node | Rule | Symbolic expression | Live substitution | Value |
|---|
EXPRESSION SETUP
Match each coefficient to the displayed factor
- Enter a and b for the powered inner factor.
- Choose the positive integer exponent p.
- Enter c and d for the second linear factor.
- Choose x0 only after defining the expression.
- Follow both branches before combining them.
WHY TWO RULES ARE NEEDED
The power is nested and the powered expression is multiplied again
The chain rule accounts for how ax+b changes inside the power. The product rule accounts for both factors changing in the multiplication.
Omitting either the inner derivative a or the second product branch produces a structurally incomplete derivative even if a special numeric example happens to hide the error.
Subject fundamentals
Five ideas that control this calculation
The expression has two outer factors
u=(ax+b)^p and v=(cx+d) are multiplied, so both can contribute to the final derivative.
The powered factor is composite
u contains an outer power and an inner affine function; differentiating it requires the chain rule multiplier a.
The product rule creates signed branches
f'=u'v+uv' preserves the sign of each evaluated contribution instead of comparing only their magnitudes.
Exponent one is still valid
When p=1 the power contribution simplifies, but the product rule remains necessary because the second linear factor still changes.
Substitution is delayed
Naming g, u, and v before inserting x0 keeps the dependency tree visible and makes a missing derivative factor easier to find.
NESTED DIFFERENTIATION
Resolve the inside function before applying product-rule addition
Let u=(ax+b)^p and v=cx+d. The chain rule differentiates u; the product rule then forms u'v+uv'.
Detailed calculation process and general formulas
g(x)=ax+bu(x)=g(x)^pu'(x)=p g(x)^(p-1) av'(x)=cf'(x)=u'(x)v(x)+u(x)v'(x)Symbols, meanings, and units
- g
- inner affine functionmodel-dependent
- u
- powered factormodel-dependent
- v
- linear factormodel-dependent
- p
- positive integer exponentdimensionless
- u'v
- product-rule branch from differentiating uoutput per x unit
- uv'
- product-rule branch from differentiating voutput per x unit
BRANCH INTERPRETATION
Contribution signs reveal reinforcement or cancellation
The final slope can be small even when both branches are large.
Powered-factor branch
-u'v carries the chain-rule multiplier.
Linear-factor branch
-uv' carries the derivative of the second factor.
Recombined slope
-The signed branch sum is the only final derivative.
Decision takeaway: Inspect branch signs before interpreting a small final slope as weak underlying change.
HAND-CHECK ROUTE
Audit the expression in this order
- Identify outer product
- Name inner function
- Differentiate inner function
- Apply power rule
- Differentiate linear factor
- Add signed branches
Applied decisions
Nested expressions where the branch view matters
Reinforcing branches
Both factors and both derivatives are positive at x0.
What the result clarifies: The tree shows two positive contributions enlarging the final slope.
Cancellation point
The powered-factor branch is positive while the linear-factor branch is negative.
What the result clarifies: A near-zero derivative can result from cancellation rather than small branch magnitudes.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Inner function
- The affine expression g(x)=ax+b nested inside the power.
- Chain rule
- The rule multiplying the outer derivative by the derivative of its inner function.
- Product rule
- The identity (uv)'=u'v+uv'.
- Derivative branch
- One signed contribution, either u'v or uv', to the product derivative.
- Evaluation point
- The entered x0 used for every intermediate substitution.
- Reconciliation
- The final check that the two displayed branches add to the reported slope.
Method references
References for this calculator's specific method
Scope and limitations
The expression family is limited to a positive integer power of one affine factor multiplied by a second affine factor. It does not parse arbitrary typed algebra.
Derivative Step-by-Step Calculator | Product and Chain Rule FAQ
Why multiply by a in the chain-rule branch?
Because the derivative of the inner function ax+b is a.
Why are there two product-rule terms?
Either factor can supply the infinitesimal change while the other remains at its current value.
Can the branches cancel?
Yes. Opposite signed contributions can produce a small or zero final derivative.
Why restrict p to an integer?
It keeps the displayed expression real for every entered x even when ax+b is negative.