Find the derivative of:y= (ax + sqrt of x^2 +b^2)^-2

[ace123]

The question is Find the derivative of:
y= (ax + sqrt of x^2 + b^2)^-2

I got this

-2(ax + sqrt of x^2 + b^2)^-3 *(a+ 2/ sqrt of x^2 + b^2) *2x

but i dno if this is right, can someone help me.

 

[PowerIso]

Yea I got the same thing minus 2/ sqrt of x^2 + b^2

It should be 1/(2*biggiantmess^1/2)

or u^1/2 = 1/(2*u^1/2)

 

[tacman]

Your solution is almost correct, but there are a few errors. Let's go through the steps together to find the correct derivative.

Step 1: Use the power rule to bring down the exponent.

y' = -2(ax + sqrt(x^2 + b^2))^(-3-1)

Step 2: Simplify the exponent.

y' = -2(ax + sqrt(x^2 + b^2))^(-4)

Step 3: Apply the chain rule to the inside function (ax + sqrt(x^2 + b^2)).

y' = -2 * (-4) * (ax + sqrt(x^2 + b^2))^(-4-1) * (a + 2x/2sqrt(x^2 + b^2))

Step 4: Simplify the expression inside the parentheses.

y' = 8(ax + sqrt(x^2 + b^2))^(-5) * (a + x/sqrt(x^2 + b^2))

Step 5: Use the power rule again to simplify the expression inside the parentheses.

y' = 8(ax + sqrt(x^2 + b^2))^(-5) * (a + x(sqrt(x^2 + b^2))^(-1))

Step 6: Simplify the expression inside the parentheses.

y' = 8(ax + sqrt(x^2 + b^2))^(-5) * (a + x/(x^2 + b^2))

Step 7: Simplify the expression further by combining like terms.

y' = 8(ax + sqrt(x^2 + b^2))^(-5) * (a(x^2 + b^2) + x)/(x^2 + b^2)

Step 8: Rewrite sqrt(x^2 + b^2) as (x^2 + b^2)^1/2.

y' = 8(ax + (x^2 + b^2)^1/2)^(-5) * (ax^2 + ab^2 + x)/(x^2 + b^2)

And there you have it! The final derivative is:

y' = 8(ax + (x^2 + b^2)^1/2)^(-5) * (ax^2 + ab^2 + x)/(x^2 + b^2)

I hope this helps clarify the process of finding the derivative. Remember to always double