Find the derivative of y=(2x²+x)²°°.
Use the chain rule. Let 2x^2 +x = u(x)
y = u^200
dy/dx = dy/du * du/dx
dy/dx = 200*(2x^2 +x)^199 *(4x+1)
To find the derivative of y=(2x²+x)²°°, we can break down the problem into several steps.
Step 1: Simplify the expression
To simplify the expression, we can expand it using the binomial theorem. Recall that the binomial theorem states that for any real numbers a and b, and a positive integer n, the expansion of (a + b)^n can be written as the sum of terms of the form C(n,r) * a^(n-r) * b^r, where C(n,r) represents the binomial coefficient.
In our case, we have (2x² + x)^2°°. Since the exponent is extremely large, it is not practical to manually expand the expression. Instead, we can apply the general rule of thumb that when the exponent is extremely large, the term that dominates the expansion is the one with the highest power.
In this case, the highest power term is 2x², and raising it to an extremely large exponent will result in terms with even higher powers of x. Therefore, all other terms in the expansion will have negligible impact on the result, and we can approximate the expression as y ≈ (2x²)²°°.
Step 2: Find the derivative of y=(2x²)²°°
To find the derivative of y=(2x²)²°°, we differentiate each term using the power rule. The power rule states that if y = ax^n, then dy/dx = nax^(n-1).
Applying the power rule to y=(2x²)²°°, we get:
dy/dx = (2°°)(2x²)^(2°°-1)
Simplifying further, we have:
dy/dx = (2x²)^(2°°-1)
Step 3: Simplify the expression
Since (2x²)^(2°°-1) is still a complex expression, let's simplify it further using logarithms. We can rewrite (2x²)^(2°°-1) as 2^[(2°°-1)log₂(2x²)]. Here, log₂ represents the logarithm to the base 2.
Using the properties of logarithms, we can simplify further:
dy/dx = 2^[(2°°-1)log₂(2x²)]
= 2^(2°°log₂(2x²) - log₂(2x²))
= 2^(2°°log₂(2x²)/log₂(2x²) - 1)
Finally, since log₂(2x²)/log₂(2x²) equals 1, the expression simplifies to:
dy/dx = 2^(2°° - 1)
Step 4: Evaluate the expression
The expression dy/dx = 2^(2°° - 1) cannot be further simplified without knowing the value of 2°°. However, given that 2 raised to an extremely large exponent is an astronomically large number, the derivative dy/dx is also an extremely large number.
Therefore, we can conclude that the derivative of y=(2x²+x)²°° is an extremely large number represented by dy/dx = 2^(2°° - 1).