Find the derivative of y=(9x^(2)-2)^(sec(x))

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Find the derivative of  y=(9x^(2)-2)^(sec(x))

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Identify Function Type: Identify the type of function and the rule needed to differentiate it. We have a composite function where an inner function (9x^2 - 2) is raised to the power of another function (sec(x)). To differentiate this, we will need to use the chain rule and the product rule, as the exponent itself is a function of x.Apply Chain Rule: Apply the chain rule to differentiate the outer function with respect to the inner function. Let u = 9x^2 - 2 and v = sec(x). Then y = u^v. The derivative of y with respect to u is v*u^(v-1) by the power rule. We will later multiply this by the derivative of u with respect to x.Differentiate Inner Function: Differentiate the inner function u = 9x^2 - 2 with respect to x. The derivative of u with respect to x is du/dx = 18x.Apply Product Rule: Apply the product rule to differentiate the outer function with respect to x. Since y = u^v, we need to differentiate v*u^(v-1) with respect to x. This requires the product rule because v is a function of x. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx).Differentiate sec(x): Differentiate v = sec(x) with respect to x. The derivative of sec(x) with respect to x is sec(x)tan(x). So dv/dx = sec(x)tan(x).Apply Product Rule with Derivatives: Apply the product rule using the derivatives from steps 3 and 5. We have u = 9x^2 - 2, du/dx = 18x, v = sec(x), and dv/dx = sec(x)tan(x). Plugging these into the product rule, we get: dy/dx = u^(v-1) * (v * du/dx + u * dv/dx)        = (9x^2 - 2)^(sec(x)-1) * (sec(x) * 18x + (9x^2 - 2) * sec(x)tan(x))Simplify Expression: Simplify the expression for dy/dx. dy/dx = (9x^2 - 2)^(sec(x)-1) * (18x * sec(x) + (9x^2 - 2) * sec(x)tan(x)) This is the simplified form of the derivative.

Find the derivative of $y=\left(9 x^{2}-2\right)^{\sec (x)}$

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Find the derivative of $y=\left(9 x^{2}-2\right)^{\sec (x)}$