Find the Derivative - d/d@VAR f(x)=3 square root of x+5

Problem

d()/d(x)*(3√(,x)+5)

Solution

1. Rewrite the radical expression using a fractional exponent to make it easier to differentiate.

√(,x)=x(1/2)

2. Apply the sum rule for derivatives, which allows for the differentiation of each term independently.

d()/d(x)*(3*x(1/2)+5)=(d(3)*x(1/2))/d(x)+d(5)/d(x)

3. Apply the power rule, d(xn)/d(x)=n*x(n−1) to the first term and the constant rule to the second term.

(d(3)*x(1/2))/d(x)=3⋅1/2*x(−1/2)

d(5)/d(x)=0

4. Simplify the expression by multiplying the coefficients and rewriting the negative exponent as a radical in the denominator.

3/2*x(−1/2)=3/(2√(,x))

Final Answer

d()/d(x)*(3√(,x)+5)=3/(2√(,x))

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