Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

Find the derivatives with respect to x of the following combinations at the given value of x.

h. √(f²(x) + g²(x)), x = 2

검증된 단계별 안내

Step 1: Recognize that the function h(x) = √(f²(x) + g²(x)) involves a composition of functions. To find h'(x), use the chain rule and the derivative of the square root function.

Step 2: The derivative of √(u) with respect to u is (1 / (2√u)). Here, u = f²(x) + g²(x). Apply this formula to h(x).

Step 3: Differentiate the inner function f²(x) + g²(x) with respect to x. Use the chain rule for each term: the derivative of f²(x) is 2f(x)f'(x), and the derivative of g²(x) is 2g(x)g'(x).

Step 4: Combine the results from Steps 2 and 3. Substitute the derivative of the inner function into the formula for h'(x): h'(x) = (1 / (2√(f²(x) + g²(x)))) * (2f(x)f'(x) + 2g(x)g'(x)).

Step 5: Evaluate h'(x) at x = 2 using the values from the table: f(2) = 8, g(2) = 2, f'(2) = 1/3, and g'(2) = -3. Substitute these values into the formula derived in Step 4 to compute h'(2).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function h(x) = f(g(x)) is h'(x) = f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate the square root function, which is a composition of the inner function f²(x) + g²(x).

Power Rule

The power rule is used to differentiate functions of the form x^n, where n is a constant. It states that the derivative of x^n is n*x^(n-1). In this context, the power rule is applied to differentiate f²(x) and g²(x), which are parts of the expression under the square root. This helps in finding the derivative of the inner function.

Sum Rule

The sum rule in calculus states that the derivative of a sum of functions is the sum of their derivatives. For the expression f²(x) + g²(x), the sum rule allows us to differentiate each term separately and then add the results. This is crucial for simplifying the derivative of the inner function before applying the chain rule.

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.Find the derivatives with respect to x of the following combinations at the given value of x.h. √(f²(x) + g²(x)), x = 2