Derivative of y= (sec5x)^((x^2)-x+2)

Question

Just need the first derivative of this function, struggling with there to begin

Answer 1

First, to simplify notation, rewrite y=f(x) ^ (g(x)), where f(x) = sec5x, g(x) = x^2-x+2

Next, take natural log (ln) of both sides:

 

ln(y) = ln(f(x)^g(x)) = g(x) * ln(f(x))

 

Now take the derivative of both sides using implicit differentiation, chain rule and the derivative of a product.

 

y'/y = g'(x)*ln(f(x)) + g(x)*f'(x)/f(x)

 

Then y' (what we want) = y*(g'(x))*ln(f(x)) + g(x)*f'(x)/f(x))

 

Finally, substitute back the function definitions:

y is your original function

g(x) and f(x) are defined above

g'(x) is the derivative of x^2 -x + 2 = 2*x-1

f'(x) is the derivative of sec(5x) = 5*tan(5x)*sec(5x)

 

Answer 2

hi there -

is this for a high school class or college?