If y = ln(5x^2 + 3), find dy/dx at x = 1.

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Question: If y = ln(5x^2 + 3), find dy/dx at x = 1.

Options:

Correct Answer: 5/8

Solution:

dy/dx = (10x)/(5x^2 + 3). At x = 1, dy/dx = (10)/(5(1) + 3) = 10/8 = 5/4.

If y = ln(5x^2 + 3), find dy/dx at x = 1.

If y = ln(5x^2 + 3), find dy/dx at x = 1.

Correct Answer: 5/4

Step 1: Identify the function y = ln(5x^2 + 3).
Step 2: Use the chain rule to find the derivative dy/dx. The derivative of ln(u) is (1/u) * (du/dx), where u = 5x^2 + 3.
Step 3: Calculate du/dx. Since u = 5x^2 + 3, the derivative du/dx = 10x.
Step 4: Substitute u and du/dx into the derivative formula: dy/dx = (1/(5x^2 + 3)) * (10x).
Step 5: Simplify the expression: dy/dx = (10x)/(5x^2 + 3).
Step 6: Now, substitute x = 1 into the derivative: dy/dx = (10(1))/(5(1)^2 + 3).
Step 7: Calculate the denominator: 5(1)^2 + 3 = 5 + 3 = 8.
Step 8: Now, calculate dy/dx at x = 1: dy/dx = 10/8.
Step 9: Simplify 10/8 to get 5/4.

Differentiation of Logarithmic Functions – The question tests the ability to differentiate a logarithmic function using the chain rule.
Application of the Chain Rule – It requires applying the chain rule correctly to find the derivative of the inner function.
Evaluation of Derivatives at a Point – The question involves evaluating the derivative at a specific value of x.