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f(x)=−(x+2)2+16f(x) = -(x+2)^2 + 16

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Full solution

Q. f(x)=−(x+2)2+16f(x) = -(x+2)^2 + 16
  1. Identify components: Identify the components of the function. The function f(x)=−(x+2)2+16f(x) = -(x + 2)^2 + 16 is a quadratic function shifted and reflected. We will use the power rule and chain rule to find the derivative.
  2. Apply rules: Apply the power rule and chain rule. The power rule states that the derivative of xnx^n is n∗x(n−1)n*x^{(n-1)}. The chain rule allows us to differentiate composite functions.
  3. Differentiate squared term: Differentiate the squared term. The derivative of (x+2)2(x + 2)^2 with respect to xx is 2⋅(x+2)2−1⋅ddx(x+2)2\cdot(x + 2)^{2-1} \cdot \frac{d}{dx}(x + 2) by the power rule and chain rule.
  4. Calculate inner function: Calculate the derivative of the inner function. The derivative of x+2x + 2 with respect to xx is 11.
  5. Combine results: Combine the results. Multiplying the derivative of the outer function by the derivative of the inner function gives us 2∗(x+2)∗12*(x + 2)*1.
  6. Apply rules: Apply the negative sign and constant rule. The derivative of a constant is 00, so the derivative of +16+16 is 00. The negative sign in front of the squared term applies to the derivative, so we have −2∗(x+2)-2*(x + 2).
  7. Simplify derivative: Simplify the derivative. The derivative of f(x)=−(x+2)2+16f(x) = -(x + 2)^2 + 16 is f′(x)=−2⋅(x+2)⋅1+0f′(x) = -2\cdot(x + 2) \cdot 1 + 0, which simplifies to f′(x)=−2x−4f′(x) = -2x - 4.

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