Find the derivative of g(x)=3cos^(-1)(x) at the point x=(1)/(4).

Question

Find the derivative of  g(x)=3cos^(-1)(x) at the point  x=(1)/(4).

Bytelearn · Accepted Answer

Identify function: Identify the function to differentiate. We are given the function g(x) = 3cos^(-1)(x), which is 3 times the inverse cosine of x. We need to find its derivative with respect to x.Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. The outer function here is 3 times a function, and the inner function is cos^(-1)(x).Differentiate outer function: Differentiate the outer function. The derivative of 3 times a function is simply 3 times the derivative of that function. So we will need to multiply the derivative of cos^(-1)(x) by 3.Differentiate inner function: Differentiate the inner function. The derivative of cos^(-1)(x) with respect to x is -1/sqrt(1-x^2). This comes from the standard derivative formulas for inverse trigonometric functions.Combine derivatives: Combine the derivatives using the chain rule. Multiplying the derivative of the outer function by the derivative of the inner function, we get g'(x) = 3 * (-1/sqrt(1-x^2)).Simplify derivative: Simplify the derivative. Simplifying the expression, we have g'(x) = -3/sqrt(1-x^2).Evaluate at x=1/4: Evaluate the derivative at the given point x = 1/4. Plugging x = 1/4 into the derivative, we get g'(1/4) = -3/sqrt(1-(1/4)^2).Calculate value: Calculate the value of the derivative at x = 1/4. Calculating the value, we have g'(1/4) = -3/sqrt(1-1/16) = -3/sqrt(15/16) = -3/sqrt(15)/4.Simplify final expression: Simplify the final expression. Simplifying the expression, we get g'(1/4) = -3/(4*sqrt(15)/4) = -3/sqrt(15).

Find the derivative of $g(x)=3 \cos ^{-1}(x)$ at the point $x=\frac{1}{4}$ .

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Find the derivative of g(x)=3cos⁡−1(x) g(x)=3 \cos ^{-1}(x) g(x)=3cos−1(x) at the point x=14 x=\frac{1}{4} x=41​.

Find the derivative of $g(x)=3 \cos ^{-1}(x)$ at the point $x=\frac{1}{4}$ .