Question

I need a real answer with solution please help.

Answer 1

A and D

Explanation:

Given the function:

`\displaystyle{f(x)=√(5-x^2)}`

In order to find the first-order derivative of square root function, we have to convert the form of surd to exponent form. Therefore:

`\displaystyle{f(x)=\left(5-x^2\right)^{(1)/(2)}}`

Since the exponent is a constant and that the base is an expression with variable, we can apply the chain rule (power rule variation):

`\displaystyle{f(x) = u^n \to f'(x) = nu^(n-1) \cdot (du)/(dx)}`

Therefore:

`\displaystyle{f'(x)=(1)/(2)\left(5-x^2\right)^{(1)/(2)-1} \cdot (d)/(dx)\left(5-x^2\right)}\\\\\displaystyle{f'(x)=(1)/(2)\left(5-x^2\right)^{(1)/(2)-(2)/(2)} \cdot (d)/(dx)\left(5-x^2\right)}\\\\\displaystyle{f'(x)=(1)/(2)\left(5-x^2\right)^{-(1)/(2)} \cdot (d)/(dx)\left(5-x^2\right)}`

Deriving 5 - x²:

`\displaystyle{(d)/(dx)\left(5-x^2\right) = 0 - 2x^(2-1)}\\\\\displaystyle{(d)/(dx)\left(5-x^2\right) = -2x}`

Thus:

`\displaystyle{f'(x)=(1)/(2)\left(5-x^2\right)^{-(1)/(2)} \cdot (-2x)}\\\\\displaystyle{f'(x)=(1)/(2)(-2x)\left(5-x^2\right)^{-(1)/(2)} }\\\\\displaystyle{f'(x)=-x\left(5-x^2\right)^{-(1)/(2)} }`

You can also convert to the square root form of:

`\displaystyle{f'(x)= (-x)/(√(5-x^2))}`

Therefore, A and D are correct.