Question

find d²y/dx² if y=√x²+5x-1​

Answer 1

Answer:

d²y/dx² = -(2x+5)/(4(x²+5x-1)^(3/2)) + 1/(x²+5x-1)^(1/2)

Step-by-step explanation:

To find d²y/dx² for y = √(x²+5x-1), we will need to use the chain rule and the product rule of differentiation.

First, we can rewrite y as:

y = (x²+5x-1)^(1/2)

Using the chain rule, we get:

dy/dx = (1/2)(x²+5x-1)^(-1/2)(2x+5)

Next, we use the product rule to find d²y/dx². We have:

d²y/dx² = d/dx(dy/dx)

= d/dx[(1/2)(x²+5x-1)^(-1/2)(2x+5)]

= [(d/dx)((1/2)(x²+5x-1)^(-1/2))(2x+5)] + [(1/2)(x²+5x-1)^(-1/2))(d/dx)(2x+5)]

Now, we need to apply the chain rule and the product rule again. We have:

(d/dx)((1/2)(x²+5x-1)^(-1/2)) = (-1/4)(x²+5x-1)^(-3/2)(2x+5)

(d/dx)(2x+5) = 2

Substituting these expressions back into the equation for d²y/dx², we get:

d²y/dx² = (-1/4)(x²+5x-1)^(-3/2)(2x+5) + [(1/2)(x²+5x-1)^(-1/2))(2)]

Simplifying this expression, we get:

d²y/dx² = -(2x+5)/(4(x²+5x-1)^(3/2)) + 1/(x²+5x-1)^(1/2)

d²y/dx² = -(2x+5)/(4(x²+5x-1)^(3/2)) + 1/(x²+5x-1)^(1/2)

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