Find the Derivative - d/dr square root of r+ fifth root of R

Problem

d()/d(r)*(√(,r)+√(5,R))

Solution

1. Rewrite the expression using rational exponents to prepare for differentiation.

√(,r)+√(5,R)=r(1/2)+R(1/5)

2. Identify the variables. Since the derivative is with respect to r the term R(1/5) is treated as a constant.

d()/d(r)*(r(1/2)+R(1/5))

3. Apply the sum rule for derivatives.

d(r(1/2))/d(r)+d(R(1/5))/d(r)

4. Differentiate the first term using the power rule, d(xn)/d(x)=n*x(n−1)

d(r(1/2))/d(r)=1/2*r(−1/2)

5. Differentiate the second term. Since R is a constant relative to r its derivative is zero.

d(R(1/5))/d(r)=0

6. Simplify the resulting expression by rewriting the negative exponent as a radical in the denominator.

1/(2√(,r))

Final Answer

d()/d(r)*(√(,r)+√(5,R))=1/(2√(,r))

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