Firstly, we need to differentiate the first function `y=(tan⁻¹(8x))²`.
Step 1: Identify the outer and inner functions
In this case, the function has the form (f(g(x)))². The outer function is 'u²' where 'u' is `(tan⁻¹(8x))` and the inner function is `g(x) = tan⁻¹(8x) = atan(8x)`.
Step 2: Apply Chain Rule
Since we have a composite function ('u' being a function of 'x'), we need to apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
The formula for the chain rule is given as:
(du/dx) = (du/dv) * (dv/dx)
Step 3: Derive the outer function
The derivative of 'u²' (with respect to 'u') is 2u, so we get 2*(tan⁻¹(8x))=2u
Step 4: Derive the inner function
The derivative of `tan⁻¹(8x)` with respect to 'x' is `8/(64x² + 1)`
Step 5: Apply chain rule
Multiply the results from steps 3 and 4 to get the derivative of the original function.
By the chain rule, we get the derivative as `2*(tan⁻¹(8x)) * (8/(64x² + 1))` = `16*atan(8x)/(64*x**2 + 1)`. Hence, that’s the derivative of the first function.
Moving forward to the second function `y=sin⁻¹(x² +1)`
Step 1: Identify the outer and inner functions
In this case, the function has the form f(g(x)) where the outer function is 'f(u) = sin⁻¹(u)' and the inner function is 'u=g(x)= x² +1'.
Step 2: Apply Chain Rule
Step 3: Derive the outer function
The derivative of `sin⁻¹(u)` (with respect to 'u') is `1/ sqrt(1 - u²)`
Step 4: Derive the inner function
The derivative of `x² +1` (with respect to 'x'), is `2x`
Step 5: Apply Chain rule
Multiply the results from steps 3 and 4 to get the derivative of the original function.
By chain rule, we would get the derivative as `(2x)/sqrt(1 - (x² + 1)²)` = `2*x/sqrt(1 - (x**2 + 1)**2)`. Hence, that’s the derivative of the second function.
Therefore, the derivatives of given functions are `16*atan(8x)/(64*x**2 + 1)` and `2*x/sqrt(1 - (x**2 + 1)**2)`.