Math  /  Calculus

QuestionIf f(x)=2ln(x+5)f(x)=2 \ln (x+5), what is the value of f(5)f^{\prime}(5) in simplest form?

Studdy Solution

STEP 1

1. We are given the function f(x)=2ln(x+5) f(x) = 2 \ln (x+5) .
2. We need to find the derivative of the function, f(x) f'(x) .
3. We will evaluate the derivative at x=5 x = 5 .

STEP 2

1. Differentiate the function f(x)=2ln(x+5) f(x) = 2 \ln (x+5) .
2. Simplify the expression for the derivative f(x) f'(x) .
3. Substitute x=5 x = 5 into the derivative to find f(5) f'(5) .

STEP 3

Differentiate the function f(x)=2ln(x+5) f(x) = 2 \ln (x+5) .
Using the chain rule, the derivative of ln(x+5) \ln (x+5) is 1x+5 \frac{1}{x+5} .
f(x)=21x+5 f'(x) = 2 \cdot \frac{1}{x+5}

STEP 4

Simplify the expression for the derivative:
f(x)=2x+5 f'(x) = \frac{2}{x+5}

STEP 5

Substitute x=5 x = 5 into the derivative:
f(5)=25+5 f'(5) = \frac{2}{5+5}
Simplify the expression:
f(5)=210 f'(5) = \frac{2}{10} f(5)=15 f'(5) = \frac{1}{5}
The value of f(5) f'(5) is:
15 \boxed{\frac{1}{5}}

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