Power
d/dx x^4
4x^3
d/dx[x(x^2+1)]
3x^2+1
d/dx (x^2+1)^3
6x(x^2+1)^2
d/dx(y)
dy/dx
Derivative meaning
Rate of change
d/dx x^5
5x^4
Product rule formula?
f'g + fg'
Chain rule idea?
Outer times inner
d/dx(y^2)
2y dy/dx
Tangent slope
Derivative
d/dx (3x^2)
6x
Quotient rule numerator?
f'g - fg'
d/dx e^(2x)
2e^(2x)
x^2+y^2=25 slope
-x/y
When to use chain rule
Composite
d/dx x^7
7x^6
d/dx[(x^2+1)/x]
(x^2-1)/x^2
Inside of (x^2+1)^3
x^2+1
Why chain rule?
y depends on x
Key first step
Identify structure
d/dx (2x^6)
12x^5
Denominator in quotient rule?
g^2
d/dx sin(3x)
3cos(3x)
Goal?
Solve for dy/dx
d/dx ln(cos2x)
-2sinx/x