Question

X=sec^4(y) find dy/dx in terms of x

Answer 1

Answer -----> dy/dx = x⁻³/⁴ / 4√( 1 - x¹/⁸ )

Given-----> x = Sec⁴y

To find ----> dy / dx , in terms of x .

Solution-----> ATQ,

x = Sec⁴ y

=> ( Sec⁴y )¹/⁴ = x¹/⁴

=> Secy = x¹/⁴

=> Sec⁻¹ ( Secy ) = Sec⁻¹ ( x¹/⁴ )

=> y = Sec⁻¹ ( x¹/⁴ )

Differentiating with respect to x , we get,

=> dy/dx = d/dx ( Sec⁻¹ x¹/⁴ )

We know that ,

d/dx ( Sec⁻¹x ) = 1 / √( 1 - x² ) , applying it here , we get,

=> dy/dx = 1 / √{1 - ( x¹/⁴)² } d/dx ( x¹/⁴)

= 1/√( 1 - x¹/⁸ ) ( 1/4 ) x¹/⁴ ⁻ ¹

= x⁻³/⁴ / 4√( 1 - x¹/⁸ )