Step-by-Step Differentiation

Subtopic: Derivatives
Topic: Calculus

Systematic Approach to Differentiation

Step 1: Identify the outermost operation — is it a sum, product, quotient, or composition?

Step 2: Apply the corresponding rule (sum rule, product rule, quotient rule, or chain rule)

Step 3: Differentiate the inner parts, applying rules recursively

Step 4: Simplify the result

Worked Example

Differentiate:

f(x)=x2*sin(3*x)

Step 1: Outermost operation is multiplication → use product rule

f(x)′=(x2)′⋅sin(3*x)+x2⋅(sin(3*x))′

Step 2: Differentiate x² → 2x. Differentiate sin(3x) using chain rule → cos(3x)·3

f(x)′=2*x*sin(3*x)+x2⋅3*cos(3*x)=2*x*sin(3*x)+3*x2*cos(3*x)

Example 2: Quotient with Chain Rule

Differentiate:

Step 1: Outermost is division → quotient rule

Step 2: Top = e^(2x), derivative = 2e^(2x) (chain rule). Bottom = x² + 1, derivative = 2x

Example 3: Multiple Chain Rules

Differentiate:

h(x)=sin3(2*x)=[sin(2*x)]3

Work from outside in: outer function is ( )³, inner is sin(2x), innermost is 2x.

h(x)′=3*[sin(2*x)]2⋅cos(2*x)⋅2=6*sin2(2*x)*cos(2*x)

Common Patterns to Recognize

Pattern: e^(f(x)) → derivative is e^(f(x)) · f'(x)

Pattern: ln(f(x)) → derivative is f'(x)/f(x)

Pattern: [f(x)]^n → derivative is n[f(x)]^(n-1) · f'(x)

Pattern: sin(f(x)) → derivative is cos(f(x)) · f'(x)

Pattern: cos(f(x)) → derivative is -sin(f(x)) · f'(x)

Verification: Always Check Your Answer

📊 More Worked Examples: Building Complexity

Example 4: Implicit function in a product

f(x)=x3⋅ln(x)

Step 1: This is a product of x³ and ln(x). Use product rule.

f(x)′=(x3)′⋅ln(x)+x3⋅(ln(x))′

=3*x2*ln(x)+x3⋅1/x=3*x2*ln(x)+x2

=x2*(3*ln(x)+1)

Example 5: Nested radicals (triple chain rule)

Label the layers: outer √, middle (1 + √), innermost (1 + x²). Work outside-in.

g(x)′=1/(2√(,1+√(,1+x2)))⋅d()/d(x)*(1+√(,1+x2))

=1/(2√(,1+√(,1+x2)))⋅1/(2√(,1+x2))⋅2*x

Example 6: Logarithmic differentiation (for variable exponents)

h(x)=xsin(x)

Neither the power rule (fixed exponent) nor exponential rule (fixed base) applies. Take ln of both sides:

ln(h)=sin(x)⋅ln(x)

Differentiate both sides (left side uses chain rule):

(h′)/h=cos(x)*ln(x)+sin(x)⋅1/x

h(x)′=xsin(x)*(cos(x)*ln(x)+sin(x)/x)

Example 7: Inverse trig with chain rule

k(x)=arctan((x−1)/(x+1))

Chain rule with arctan (derivative 1/(1+u²)) and quotient rule inside:

Surprise! This simplifies beautifully. (In fact, arctan((x-1)/(x+1)) = arctan(x) - π/4.)

🎯 Decision Tree: Which Rule to Use?

Ask these questions in order:

1. Can I simplify first? (Algebraic manipulation, log properties, trig identities)

2. What's the OUTERMOST operation?
   • Sum/difference → Sum rule (differentiate term by term)
   • Product → Product rule
   • Quotient → Quotient rule (or rewrite as product with negative exponent)
   • Composition f(g(x)) → Chain rule

3. Work inward, applying rules recursively until you reach basic functions (x, sin x, eˣ, etc.)

⚠️ Common Process Mistakes

Starting with the innermost function instead of outermost.
Wrong approach for sin(x²): "First I'll differentiate x² to get 2x..."
Right approach: "Outer is sin, inner is x². Derivative of outer at inner × derivative of inner."

Confusing function and variable.
In eˣ², think: "This is e^(something), not (e^x)^2." The something is x², so chain rule gives eˣ² · 2x.

Incomplete chain rule.
Every function inside another function needs its derivative multiplied. sin(x³) has derivative cos(x³)·3x², not cos(x³).

💡 Simplification Tricks

• Factor before differentiating: x³ - x = x(x² - 1) might be easier as a product

• Expand if the product is simple: (x+1)² = x² + 2x + 1 avoids product rule

• Use log properties: ln(x²y³) = 2ln(x) + 3ln(y) splits into simpler pieces

• Rewrite radicals as powers: √x = x^(1/2), ∛x = x^(1/3) for power rule

• Simplify after differentiating: Factor out common terms, combine fractions

🔗 Connection: Differentiation Sets Up Integration

Every differentiation problem teaches you an integration problem backwards. When you prove that d/dx[x·sin(x)] = sin(x) + x·cos(x), you've also shown that ∫(sin(x) + x·cos(x))dx = x·sin(x) + C.

Keep a "derivative table" in your head — it doubles as an "integral table" when read backwards!