Q&A for How to Differentiate the Square Root of X

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How do I use the chain rule?

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For the equation in the article title (y = √x), you don't need to use the chain rule, as there is not a function within a function. An example of a function that requires use of the chain rule for differentiation is y = (x^2 + 1)^7. To solve this, make u = x^2 + 1, then substitute this into the original equation so you get y = u^7. Differentiate u = x^2 + 1 with respect to x to get du/dx = 2x and differentiate y = u^7 with respect to u to get dy/du = 7u^6. Multiply dy/du by du/dx to cancel out the du and get dy/dx = 7u^6 * 2x = 14x * u^6. Substitute u = x^2 + 1 into dy/dx = 14x * u^6 to get your answer, which is dy/dx = 14x(x^2 + 1)^6.

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Question

How do I differentiate √x-1 using the first principle?

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Since the outer function is sqrt(x), you rewrite sqrt(x-1) as (x-1)^(1/2), and differentiating using the power rule gives you 1/2*(x-1)^(1/2-1)=1/2*(x-1)^(-1/2)=1/(2*sqrt(x-1)). You would normally use the chain rule for compositions: the derivative of the inner function, x-1, is 1. 1 multiplying by anything won't change anything, so your answer may be anything equivalent to 1/(2*sqrt(x-1)).

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What is the derivative of y=√(x+4) by the power rule?

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Rewrite the square root of (x + 4) as (x + 4)^1/2. Then use the power rule. Bring the 1/2 to the front, and subtract 1 from the exponent. Thus, it's (1/2)(x+4)^(-1/2), which equals 1 / [2√(x+4)]. .

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How do I find the derivative of the square root of x using the first principle?

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The derivative of the square root of x is 1/(2 times the square root of x).

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How do I find dY/dX if Y = (1 - √X)(1 + √X)?

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Y simplifies to 1 - X. Therefore, dY/dX is the derivative of 1, which is 0, plus the derivative of -X, which is -1. Thus, dY/dX = -1.

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