How to Differentiate a Power to a Bracket

Differentiating a power to a bracket is a common mathematical operation that can be performed using the chain rule. The chain rule states that if you have a function of the form f(g(x)), then the derivative of f with respect to x is given by:

f'(x) = f'(g(x)) * g'(x)

In this case, we have a power function, which is of the form f(x) = x^n. The derivative of a power function is given by:

f'(x) = n * x^(n-1)

And we have a bracket function, which is of the form g(x) = (ax + b). The derivative of a bracket function is given by:

g'(x) = a

So, to differentiate a power to a bracket, we can use the chain rule as follows:

(x^n)' = n * x^(n-1) * (ax + b)'

 = n * x^(n-1) * a

 = an * x^(n-1)

Therefore, the derivative of a power to a bracket is the power of the outer function multiplied by the derivative of the outer function multiplied by the derivative of the inner function.

Here are some examples of how to differentiate a power to a bracket:

(x^3)’ = 3 * x^2
(((2x + 1)^5)’ = 5 * (2x + 1)^4 * 2 = 10 * (2x + 1)^4
((x^2 – 3x + 5)^4)’ = 4 * (x^2 – 3x + 5)^3 * (2x – 3) = 8x * (x^2 – 3x + 5)^3

Conclusion

Differentiating a power to a bracket is a straightforward process that can be performed using the chain rule. By understanding the steps involved, you can differentiate any power to a bracket expression with ease.

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