multiple derivativesdetermine first,second derivative of y= e^6x-(lnx)/x
We have the expression y = e^6x - (ln x)/x
The first derivative is 6*e^6x - [x^-1/x - ln x/x^2]
=> 6*e^6x - 1/x^2 + (ln x)/x^2
=> 6*e^6x + (ln x - 1)/x^2
The second derivative is:
36*e^6x + (1/x)/x^2 - 2*(ln x - 1)/x^3
=> 36*e^6x + 1/x^3 - 2*(ln x - 1)/x^3
=> 36*e^6x + (3 - 2*ln x)/x^3
To determine the second derivative, we'll have to determine the 1st derivative, for the beginning. We'll differentiate y with respect to x.
dy/dx = d(e^6x+(lnx)/x)/dx
We'll apply chain rule for the first term of the sum and the quotient rule for the 2nd terms of the sum.
dy/dx = 6e^6x + [(lnx)'*x - (lnx)*(x)']/x^2
dy/dx =6e^6x+ (x/x - lnx)/x^2
dy/dx = 6e^6x + (1 - lnx)/x^2
dy/dx = 6e^6x + (lne - lnx)/x^2
dy/dx = 6e^6x + [ln(e/x)]/x^2
Now, we'll determine the second derivative:
d^2y/dx^2 = [d[6e^5x + [ln(e/x)]/x^2/dx
In other words, we'll determine the derivative of the expression of the 1st derivative.