Power Rule of Derivatives
The power rule is a fundamental rule in calculus used to find the derivative of a function that is in the form of a power function
The power rule is a fundamental rule in calculus used to find the derivative of a function that is in the form of a power function. It helps us determine how the rate of change of a function depends on its exponent.
The power rule states that if we have a function of the form:
f(x) = x^n
Where x is the independent variable and n is a constant (exponent), then the derivative of the function is given by:
f'(x) = n*x^(n-1)
In other words, to find the derivative of a power function, we decrease the exponent by 1 and multiply it by the coefficient of x.
For example, let’s find the derivative of the function f(x) = 3x^4. By applying the power rule, we have:
f'(x) = 4 * 3x^(4-1)
= 12x^3
So, the derivative of f(x) = 3x^4 is f'(x) = 12x^3. This means that the rate at which the function is changing with respect to x is given by 12x^3.
The power rule is incredibly useful because it allows us to find the derivatives of various types of functions. We can extend the power rule to include more complicated functions by applying it repeatedly or by combining it with other rules of differentiation, such as the product rule or the chain rule.
It is important to note that the power rule does not apply if the exponent is not a constant or if the base of the power is a function of x. In such cases, alternative rules need to be applied, such as the exponential or logarithmic rules of differentiation.
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