Find the Slope x=8

Problem

y=√(3,x)

Solution

1. Identify the function to be differentiated, which is y=√(3,x)

2. Rewrite the radical expression as a power to make differentiation easier using the rule √(n,x)=x1/n

y=x1/3

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

d(y)/d(x)=1/3*x(1/3−1)

4. Simplify the exponent.

d(y)/d(x)=1/3*x(−2/3)

5. Rewrite the expression using positive exponents and radical notation.

d(y)/d(x)=1/(3√(3,x2))

6. Substitute the given value x=8 into the derivative to find the slope of the tangent line at that point.

d(y)/d(x)=1/(3√(3,8))

7. Evaluate the numerical expression by calculating 8=64 and then √(3,64)=4

d(y)/d(x)=1/(3⋅4)

8. Calculate the final value.

d(y)/d(x)=1/12

Final Answer

d(√(3,x))/d(x)* at *x=8=1/12

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