Find the derivative
** (sq root)
y = (square root) x + (2nd sq root) x +
(3rd sq root) x
** All the x's is under three sq roots
y' = 1/2(sq rt) x + 1/(sq rt) x +
3/2(sq rt) x
Is this right?
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If square root) x + (2nd sq root) x +
(3rd sq root) x
mean
sqrt(x )+2*sqrt(x)+3*sqrt(x)
type:
derivative sqrt(x )+ 2* sqrt(x)+3*sqrt(x)
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To find the derivative of the given function, we can use the power rule for differentiation. However, since the function contains square roots nested within each other, we need to simplify it before differentiating.
Let's rewrite the function in a simpler form:
y = (√x)^1 + (√x)^2 + (√x)^3
To simplify this, we can combine the exponents:
y = (√x + (√x)^2 + (√x)^3
Now we can differentiate the simplified function using the power rule. The power rule states that if we have a function of the form f(x) = x^n, then the derivative f'(x) = nx^(n-1).
Differentiating each term separately:
For the first term, we have (√x + (√x)^2 + (√x)^3, which can be rewritten as (x^(1/2) + x + x^(3/2)).
To differentiate the first term, we multiply the exponent by the coefficient in front:
d/dx (x^(1/2)) = 1/2 * x^(-1/2) = (1/2) / (√x) = 1/(2√x)
Similarly, differentiate the second term:
d/dx (x) = 1
Differentiate the third term:
d/dx (x^(3/2)) = (3/2) * x^(1/2) = (3/2) * (√x) = 3/(2√x)
Now, we can rewrite the derivative by putting these terms together:
y' = 1/(2√x) + 1 + 3/(2√x)
Therefore, the derivative of the given function should be:
y' = 1/(2√x) + 1 + 3/(2√x)
Based on the simplification and application of the power rule, the expression you provided for the derivative is correct:
y' = 1/2(√x) + 1/(√x) + 3/2(√x)