Use differentiation to verify each equation.
d/dx(x / √1...

Question

Use differentiation to verify each equation.

d/dx(x / √1−x²) = 1 / (1−x²)^3/2.

Accepted Answer

Welcome back, everyone. Calculate the derivative of the function F of X equals 3 x divided by square root of 1 minus 3 x 2. We're given 4 answer choices A says -3 divided by 1 minus 3x2 raise to the power of three halves, B 3 divided by 1 minus 3X2 raise the power of three halves C. 3 divided by square root of 1 minus 3 X2 and D3 divided by 1 minus 3 x 2 squad. So we're given F of X and we want to evaluate F of X, the derivative of F of X, right? And essentially we're given a rational function, so we have to apply the quotient rule. Let's remember the quotient rule, it says that if we want to find The derivative of F divided by G we take F multiplied by G, subtract FG, and divided by g squared. And in this case, well, we can essentially say that our top function is F and our bottom function is G, right? So essentially what we want to do is Evaluate the derivative of 3 acts. Multiply it by the bottom function which is square root of 1 minus 3 x 2, and we want to subtract 3 X which is multiplied by the derivative of square root of 1 minus 3 x 2, right? And then we are dividing that by square root. Off 1 minus 3 x 2 squad right because we have to squareg. So we have the required expression and now we can evaluate the derivatives. So the derivative of 3 X based on the power rule is 3 multiplied by the derivative of X, which is 3 multiplied by 1 or simply 3. And now let's value the derivative of square root of 1 minus 3 x 2. Because we have a radical, we can rewrite it as 1 minus 3X2 raise the power of 1/2 because we want to apply the power rule, right? And according to the power rule, we bring down the exponents, so that's 1/2 multiplied by 1 minus 3X2d. We're decreasing the original exponent by 1 unit, which gives us -12, but the inner function. Does not correspond to just X, right? We have a polynomial, so we have to apply the chain rule and multiply by the derivative of 1 minus 3 X squad. So now what do we get? Well, essentially, we can rewrite it as 1 divided by 2 square root of 1 minus 3 X 2 because we have an exponent of 1/2 so we can write our term in the denominator with square root, right? And then for the remaining derivative, the derivative of 1 is 0. And the derivative of -3 X squared is -6X. We're applying the power rule once again. So now what do we get? Well, -6 divided by 2 gives us -3 X. And then in the denominator, we still have square root of 1 minus 3X2d. So we have all the components that we need and we can say that F of X is equal to. 3 multiplied by square root of 1 minus 3 x 2 minus 3 x multiplied by. 3 x divided by square root of 1 minus 3 x 2. And this is divided by. The bottom function squared. So if we square square root, we get 1 minus 3 x 2. And now we have all of the required derivatives. We just want to simplify this result. We can essentially find the common denominator, so let's multiply the first term by square root of 1 minus 3 X2, right? And for simplicity, we can rewrite it in an exponential form. So we're going to get. Our fraction, let's begin with the denominator 1 minus 3 X2, and now our common denominator is going to be 1. Minus 3x2 raise the power of 1/2 because that square root of 1 minus 3 x 2. And now for the first term, we're going to have 3 and 1 minus 3 X squad because we're squaring the radical term. Plus 3 X multiplied by 3 X because we have minus minus, right? So we can say plus 9 X squared. And now what we're going to do is simply understand that we have a double division, so 12 plus 1 gives us an exponent of 3 halves in the denominator. 1 minus 3 x 2 raise the power of 3 halves, and in the numerator we're going to have 3. Minus 9 x2 + 9 x2, so just 3 in the numerator, and this means that we have the final answer. Now considering the answer choices, we can conclude that the correct answer to this problem corresponds to the answer choice B. Thank you for watching.

Use differentiation to verify each equation.
d/dx(x / √1−x²) = 1 / (1−x²)^3/2.

Key Concepts

Differentiation

Chain Rule

Simplifying Derivatives

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