Find the Derivative - d/d@VAR f(x)=e^( square root of x+21)+e^p

Problem

d()/d(x)*(e^√(,x+21)+e^p)

Solution

1. Identify the components of the function ƒ(x)=e^√(,x+21)+e^p noting that e^p is a constant with respect to x

2. Apply the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives.

d(ƒ(x))/d(x)=d(e^√(,x+21))/d(x)+d(e^p)/d(x)

3. Differentiate the constant term e^p Since it does not contain the variable x its derivative is zero.

d(e^p)/d(x)=0

4. Apply the chain rule to the first term e^√(,x+21) The derivative of e^u is e^u⋅d(u)/d(x) where u=√(,x+21)

d(e^√(,x+21))/d(x)=e^√(,x+21)⋅d(√(,x+21))/d(x)

5. Differentiate the inner function √(,x+21) using the power rule, where √(,x+21)=(x+21)^(1/2)

d(x+21)/d(x)=1/2*(x+21)^(−1/2)⋅d(x+21)/d(x)

6. Simplify the expression by noting that d(x+21)/d(x)=1

d(√(,x+21))/d(x)=1/(2√(,x+21))

7. Combine the results to find the final derivative.

d(ƒ(x))/d(x)=e^√(,x+21)⋅1/(2√(,x+21))

Final Answer

d()/d(x)*(e^√(,x+21)+e^p)=(e^√(,x+21))/(2√(,x+21))

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