The Product Rule
What if you need to differentiate two functions multiplied together? You can't just multiply the derivatives. You need the Product Rule.
The Product Rule
d/dx[f·g] = f'g + fg' - derivative of first times second plus first times derivative of second
Illustration: product-rule
Product Rule
Product Rule
d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x)
Derivative of first times second, plus first times derivative of second
Basic Product Rule
Medium
Find d/dx[x^2 * sin(x)]
1
Identify f and g
f(x) = x^2, g(x) = sin(x)
2
Find derivatives
f'(x) = 2x, g'(x) = cos(x)
3
Apply product rule
= f'g + fg'
= 2x*sin(x) + x^2*cos(x)
Answer:
2x*sin(x) + x^2*cos(x)
Practice
1. d/dx[x*e^x] = ?
f=x, g=e^x. f'=1, g'=e^x. Product rule: 1*e^x + x*e^x = e^x + xe^x
Key Takeaways
Product Rule: (fg)' = f'g + fg'
NOT just f' times g'!
Remember: 'first dee-second plus second dee-first'