Find the Derivative - d/dx y = log base 5 of square root of x^2-1

Problem

d()/d(x)*(log_5)(√(,x2−1))

Solution

1. Rewrite the expression using the properties of logarithms and exponents to simplify the differentiation process.

y=(log_5)((x2−1)(1/2))

2. Apply the power rule for logarithms, (log_b)(an)=n*(log_b)(a) to move the exponent in front of the log.

y=1/2*(log_5)(x2−1)

3. Use the change of base formula, (log_b)(u)=ln(u)/ln(b) to rewrite the expression in terms of the natural logarithm.

y=1/(2*ln(5))*ln(x2−1)

4. Differentiate the expression using the chain rule, where d(ln(u))/d(x)=1/u⋅d(u)/d(x)

d(y)/d(x)=1/(2*ln(5))⋅1/(x2−1)⋅d(x2−1)/d(x)

5. Calculate the derivative of the inner function x2−1 which is 2*x

d(y)/d(x)=1/(2*ln(5))⋅1/(x2−1)⋅2*x

6. Simplify the resulting expression by canceling the constant factor of 2.

d(y)/d(x)=x/((x2−1)*ln(5))

Final Answer

d((log_5)(√(,x2−1)))/d(x)=x/((x2−1)*ln(5))

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