Evaluate the Derivative at x=0 y = square root of 6+3x , x=0

Problem

d(√(,6+3*x))/d(x),x=0

Solution

1. Rewrite the square root function as a power to make it easier to differentiate.

y=(6+3*x)(1/2)

2. Apply the chain rule by differentiating the outer power and multiplying by the derivative of the inner expression 6+3*x

d(y)/d(x)=1/2*(6+3*x)(−1/2)⋅d(6+3*x)/d(x)

3. Differentiate the inner expression, where the derivative of 6 is 0 and the derivative of 3*x is 3

d(y)/d(x)=1/2*(6+3*x)(−1/2)⋅3

4. Simplify the derivative expression by moving the negative exponent to the denominator.

d(y)/d(x)=3/(2√(,6+3*x))

5. Substitute x=0 into the derivative to find the value at that specific point.

d(y)/d(x)|=3/(2√(,6+3*(0)))

6. Evaluate the resulting numerical expression.

d(y)/d(x)|=3/(2√(,6))

7. Rationalize the denominator if necessary.

d(y)/d(x)|=(3√(,6))/12

d(y)/d(x)|=√(,6)/4

Final Answer

d(√(,6+3*x))/d(x)|=√(,6)/4

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