Answer to Question #102275 in Calculus for Nimra

Question

f(x,y)=tanx².secx²(x+6x+2xy). Compute partial derivatives of f with respect to x, y.

Answer 1

The function is given by:

f(x,y) = tan (x²) sec(x²) (x+6x+2xy)

= tan (x²) sec(x²) (7x+2xy)

Now:

The partial derivative of f(x,y) with respect to x is:

f_x = tan(x²)sec(x²) (7 + 2y) + (7x + 2xy)[ sec (x²) (2x sec² (x²)) + tan (x²) (2x sec(x²) tan(x² ))] [ By using Product Rule of Derivative ]

= tan(x²)sec(x²) (7 + 2y) + x(7 + 2y)[ 2x sec³ (x²) + 2x tan² (x²) sec(x²) ]

= (7 + 2y) sec(x²) ( tan(x²) + 2x² sec² (x²) + 2x² tan² (x²) )

And:

The partial derivative of f(x,y) with respect to y is:

f_y = tan(x²)sec(x²) (0 + 2x) + (7x + 2xy) (0) [ By using Product Rule of Derivative and the partial derivative of tan(x²)sec(x²) with respect to y is zero. ]

= 2x tan(x²)sec(x²)