Find the Derivative - d/dx e^( square root of 3x)

Problem

d(e√(,3*x))/d(x)

Solution

1. Identify the outer and inner functions to apply the chain rule, where the outer function is eu and the inner function is u=√(,3*x)

2. Apply the chain rule for the exponential function, which states that d(eu)/d(x)=eu⋅d(u)/d(x)

d(e√(,3*x))/d(x)=e√(,3*x)⋅d(√(,3*x))/d(x)

3. Rewrite the square root as a fractional power to prepare for the power rule.

√(,3*x)=(3*x)(1/2)

4. Apply the chain rule again to differentiate (3*x)(1/2) multiplying the derivative of the outer power by the derivative of the inner linear term 3*x

d(3*x)/d(x)=1/2*(3*x)(−1/2)⋅3

5. Simplify the expression by combining the constants and moving the negative exponent to the denominator.

3/(2√(,3*x))

6. Combine all parts to find the final derivative.

e√(,3*x)⋅3/(2√(,3*x))

Final Answer

d(e√(,3*x))/d(x)=(3*e√(,3*x))/(2√(,3*x))

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