Math-Linux.com

Knowledge base dedicated to Linux and applied mathematics.

Home > Mathematics > Derivative of a function > Derivative of u*v , u times v

Derivative of u*v , u times v

All the versions of this article: <English> <français>

The derivative of u(x).v(x) is given by : u’(x).v(x) + u(x). v’(x). Let’s prove it using limits.

Derivative of u(x) * v(x)

Let $u(x)$ and $v(x)$ be two functions of the real variable $x$. The derivative $f’(x)$ of the function $f(x) = u(x) \cdot v(x)$ is given by:

$$ \forall x \in \mathbb{R}, \quad f’(x) = u’(x) \cdot v(x) + u(x) \cdot v’(x) $$

Proof

Using the definition of the derivative, we have:

$$ \begin{aligned} f’(x) &= \lim_{h\to 0} \frac{u(x+h)v(x+h) - u(x)v(x)}{h} \\ &= \lim_{h\to 0} \frac{u(x+h)v(x+h) - u(x)v(x+h) + u(x)v(x+h) - u(x)v(x)}{h} \\ &= \lim_{h\to 0} \frac{v(x+h)(u(x+h)-u(x))}{h} + \lim_{h\to 0} \frac{u(x)(v(x+h)-v(x))}{h} \\ &= v(x) \cdot \lim_{h\to 0} \frac{u(x+h)-u(x)}{h} + u(x) \cdot \lim_{h\to 0} \frac{v(x+h)-v(x)}{h} \\ &= u’(x) \cdot v(x) + u(x) \cdot v’(x) \end{aligned} $$

Therefore, we have:

$$ \forall x \in \mathbb{R}, \quad f’(x) = u’(x) \cdot v(x) + u(x) \cdot v’(x) $$

Also in this section

  1. Chain rule proof - derivative of a composite function
  2. Derivative of 1/x
  3. Derivative of arccos x
  4. Derivative of arcsin x
  5. Derivative of arctan x
  6. Derivative of cos x
  7. Derivative of exp x, e^x
  8. Derivative of exp(u) , exp(u(x))
  9. Derivative of inverse functions
  10. Derivative of ln u
  11. Derivative of ln x
  12. Derivative of sin x
  13. Derivative of square root of x
  14. Derivative of tan x
  15. Derivative of u*v , u times v
  16. Derivative of u/v
  17. Derivative of x power n
  18. Derviative of 1/u