Expert: Ahmed Salami - 8/1/2009

Question
Hi

I know how to take a derivative of a straightforward function with only one variable, but can you please show me how to take a derivative of an equation with 2 variables where one of the variables is also an exponent?

For example,
(-100x^2 + 1000x + 10)* ((1+y)^x)

I only need the derivative with respect to x.

Thanks,
Grace

Answer
Hi Grace,
The key to finding the derivative of a function involving variables x and y with respect to x is to add a 'dy/dx' after differentiating the y expression. Now, i'll show you how to find the derivative of (1+y)^x with respect to x and you can then combine it with (-100x^2 + 1000x + 10) using the product rule.
Let z = (1+y)^x
taking natural logarithm of both sides
ln z = ln (1+y)^x
ln z = x.ln(1+y)
differentiating both sides with respect to x
(1/z)dz/dx = x.(1/1+y)dy/dx  +  1.ln(1+y)
(1/z)dz/dx = (x/1+y)dy/dx  +  ln(1+y)
dz/dx = z[(x/1+y)dy/dx  +  ln(1+y)]
dz/dx = (1+y)^x[(x/1+y)dy/dx  +  ln(1+y)]
which is the derivative of (1+y)^x with respect to x

Regards