Find the Slope x=4

Problem

y=√(,x),x=4

Solution

1. Identify the function to be differentiated. The task asks for the slope of the square root function, y=√(,x) at the specific point x=4

2. Rewrite the function using a fractional exponent to make differentiation easier.

y=x(1/2)

3. Apply the power rule for derivatives, which states that d(xn)/d(x)=n*x(n−1)

d(y)/d(x)=1/2*x(−1/2)

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

d(y)/d(x)=1/(2√(,x))

5. Substitute the given value x=4 into the derivative to find the slope at that point.

m=1/(2√(,4))

6. Calculate the final numerical value.

m=1/(2⋅2)

m=1/4

Final Answer

d(√(,x))/d(x)|=1/4

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