How do you use quotientrule to find the slope tangent to a curve?

Answer 1

The quotient rule tells says that:

If: f(x) = g(x) / h(x)

Then: f'(x) = (g'(x) · h(x) - g(x) · h'(x)) / h(x)2

That derivative gives you the slope of the original curve's tangent at any given x-coordinate.

For example, if:

f(x) = axn / bxm

then:

f'(x) = (anxn-1 · bxm - axn · bmxm - 1) / b2x2m

This of course simplifies considerably, due to the simplicity of the original function:

= (abnxm+n-1 - abmxm+n-1) / b2x2m

= (n - m)abxm+n-1 / b2x2m

= (n - m)axn-m-1 / b

= (n - m)(a/b)xn-m-1

Which makes perfect sense if you rearrange our initial function and take it's derivative in a simpler manner:

f(x) = axn / bxm

f(x) = (a/b)xn-m

f'(x) = (n-m)(a/b)xn-m-1