Using Black Scholes, if the Call Price, Stock price, SD, Rf rate and Time to exp. for an option are known, is there a closed form solution to calculate the implied Strike price.

If not, in using Newton's method to do the calculation is there a formula to calculte the first derivative of the call price with respect to the strike price.

Given that Delta (1st derivative of call wrt stock price) = N((ln(S/X)+(b+sd^2/2)*T)/(sd*sqrt(T)), I thought that it might = N((ln(X/S)+(b+sd^2/2)*T)/(sd*sqrt(T)). It is close but not quite.