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Derivative function of rootx rootx?
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Derivative function of rootx rootx?
Derivative of the function √x * √x
To find the derivative of the function √x * √x, we can use the product rule. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product u(x) * v(x) is given by the formula:
(u(x) * v(x))' = u'(x) * v(x) + u(x) * v'(x)
In this case, u(x) = √x and v(x) = √x. Let's find the derivatives of these functions first:
To find the derivative of √x, we can rewrite it as x^(1/2) and then apply the power rule. The power rule states that if we have a function of the form x^n, where n is a constant, then the derivative is given by:
d/dx (x^n) = n * x^(n-1)
Using the power rule, the derivative of x^(1/2) is:
d/dx (x^(1/2)) = (1/2) * x^(-1/2)
Simplifying further, we get:
d/dx (√x) = (1/2) * x^(-1/2)
The derivative of √x is the same as the derivative of x^(1/2), which we have already found to be (1/2) * x^(-1/2).
Now, let's apply the product rule to find the derivative of √x * √x:
(√x * √x)' = (√x)' * √x + √x * (√x)'
Substituting the derivatives we found earlier:
(√x * √x)' = ((1/2) * x^(-1/2)) * √x + √x * ((1/2) * x^(-1/2))
Now, let's simplify this expression:
(√x * √x)' = (1/2) * x^(-1/2) * √x + √x * (1/2) * x^(-1/2)
Using the properties of exponents, we can rewrite x^(-1/2) as 1/√x:
(√x * √x)' = (1/2) * √x * (1/√x) * √x + √x * (1/2) * (1/√x)
Simplifying further:
(√x * √x)' = (1/2) * √x * √x/√x + √x * (1/2) * 1/√x
(√x * √x)' = (1/2) * √x + √x * (1/2) * 1/√x
(√x * √x)' = (1/2) * √x + (1/2) * √x
(√x * √x)' = √x
Conclusion:
The derivative of the function
To find the derivative of the function √x * √x, we can use the product rule. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product u(x) * v(x) is given by the formula:
(u(x) * v(x))' = u'(x) * v(x) + u(x) * v'(x)
In this case, u(x) = √x and v(x) = √x. Let's find the derivatives of these functions first:
To find the derivative of √x, we can rewrite it as x^(1/2) and then apply the power rule. The power rule states that if we have a function of the form x^n, where n is a constant, then the derivative is given by:
d/dx (x^n) = n * x^(n-1)
Using the power rule, the derivative of x^(1/2) is:
d/dx (x^(1/2)) = (1/2) * x^(-1/2)
Simplifying further, we get:
d/dx (√x) = (1/2) * x^(-1/2)
The derivative of √x is the same as the derivative of x^(1/2), which we have already found to be (1/2) * x^(-1/2).
Now, let's apply the product rule to find the derivative of √x * √x:
(√x * √x)' = (√x)' * √x + √x * (√x)'
Substituting the derivatives we found earlier:
(√x * √x)' = ((1/2) * x^(-1/2)) * √x + √x * ((1/2) * x^(-1/2))
Now, let's simplify this expression:
(√x * √x)' = (1/2) * x^(-1/2) * √x + √x * (1/2) * x^(-1/2)
Using the properties of exponents, we can rewrite x^(-1/2) as 1/√x:
(√x * √x)' = (1/2) * √x * (1/√x) * √x + √x * (1/2) * (1/√x)
Simplifying further:
(√x * √x)' = (1/2) * √x * √x/√x + √x * (1/2) * 1/√x
(√x * √x)' = (1/2) * √x + √x * (1/2) * 1/√x
(√x * √x)' = (1/2) * √x + (1/2) * √x
(√x * √x)' = √x
Conclusion:
The derivative of the function
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Derivative function of rootx rootx?
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Derivative function of rootx rootx? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about Derivative function of rootx rootx? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Derivative function of rootx rootx?.
Derivative function of rootx rootx? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about Derivative function of rootx rootx? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Derivative function of rootx rootx?.
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