This lesson covers two important differentiation techniques:
• Chain Rule — used when a function is inside another function (composite functions).
• Implicit Differentiation — used when x and y are mixed together in an equation.

Throughout this page, we use dy/dx for derivatives and ^ for exponents.

The Chain Rule helps us differentiate composite functions — functions inside other functions.

General formula:

y = f(g(x))
dy/dx = f'(g(x)) * g'(x)

Chain Rule Examples (Click to Expand)

Example 1
y = (3x^2 + 5)^7
Outer derivative:
7(3x^2 + 5)^6
Inner derivative:
6x
Final:
dy/dx = 42x(3x^2 + 5)^6

---

Example 2
y = sqrt(x^4 - 2x)
Rewrite:
y = (x^4 - 2x)^(1/2)
Derivative:
dy/dx = (1/2)(x^4 - 2x)^(-1/2) * (4x^3 - 2)
Simplified:
dy/dx = (4x^3 - 2)/(2sqrt(x^4 - 2x))

---

Example 3
y = sin(5x^3)
Outer derivative:
cos(5x^3)
Inner derivative:
15x^2
Final:
dy/dx = 15x^2 cos(5x^3)

---

Example 4
y = (cos(4x))^2
dy/dx = 2cos(4x) * (-sin(4x)) * 4
Final:
dy/dx = -8cos(4x)sin(4x)

---

Example 5
y = (x^2 + 10)^(-3)
Derivative:
dy/dx = -6x(x^2 + 10)^(-4)

Implicit equations mix x and y together, like x^2 + y^2 = 25. We cannot isolate y easily, so we differentiate both sides with respect to x.

Whenever we differentiate a term containing y, we multiply by dy/dx because y depends on x.

Implicit Differentiation Examples (Click to Expand)

Example 1
x^2 + y^2 = 25
Derivative:
2x + 2y(dy/dx) = 0
Solve:
dy/dx = -x / y

---

Example 2
x^2 + xy + y^2 = 10
Derivative:
2x + x(dy/dx) + y + 2y(dy/dx) = 0
Group terms:
(x + 2y)dy/dx = -(2x + y)
Final:
dy/dx = -(2x + y)/(x + 2y)

---

Example 3
xy = 5
Derivative:
x(dy/dx) + y = 0
Solve:
dy/dx = -y/x

Chain Rule

1. y = (2x^3 + 1)^5
2. y = sqrt(x^2 + 4x)
3. y = sin(3x^2)
4. y = (cos x)^4
5. y = (x^3 + 7)^(-2)

Implicit Differentiation

1. x^2 + y^2 = 16
2. x^2 + xy = 8
3. x^2y + y = 3
4. x^3 + y^3 = 9
5. xy + y^2 = 10