y = 3(4 + 3x^2)^15
y’ = 90x(4 + 3x^2)^14
To answer this question, we can use the chain rule of differentiation, which states that if we have a function f(g(x)) then,
(f(g(x)))’ = f'(g(x)) * g'(x)
In this case, we have a function y that involves an inner function u = 4 + 3x^2 raised to the power of 15. So, we can take u as the base and apply the power rule of differentiation. Then, we can multiply it by the derivative of the inner function.
Step-by-step solution:
1. Let u = 4 + 3x^2, then y = 3u^15
2. Find the derivative of u with respect to x, which is du/dx = 6x
3. Apply the power rule of differentiation: d/dx (u^n) = n * u^(n-1) * du/dx
4. Substitute n = 15 and du/dx = 6x: d/dx (u^15) = 15 * u^14 * 6x = 90x * (4 + 3x^2)^14
5. Multiply the result by the constant factor 3 from the original function: dy/dx = 3 * 90x * (4 + 3x^2)^14
6. Simplify and rewrite the final result: dy/dx = 270x * (4 + 3x^2)^14
So, the derivative of y with respect to x is 270x * (4 + 3x^2)^14. This represents the instantaneous rate of change of y with respect to x at any given point.
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