Identify components: Identify the components of the function. The function f(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. Apply rules: Apply the power rule and chain rule. The power rule states that the derivative of x^n is n*x^(n-1). The chain rule allows us to differentiate composite functions. Differentiate squared term: Differentiate the squared term. The derivative of (x + 2)^2 with respect to x is 2*(x + 2)^(2-1) * d/dx(x + 2) by the power rule and chain rule. Calculate inner function: Calculate the derivative of the inner function. The derivative of x + 2 with respect to x is 1. Combine results: Combine the results. Multiplying the derivative of the outer function by the derivative of the inner function gives us 2*(x + 2)*1. Apply rules: Apply the negative sign and constant rule. The derivative of a constant is 0, so the derivative of +16 is 0. The negative sign in front of the squared term applies to the derivative, so we have -2*(x + 2). Simplify derivative: Simplify the derivative. The derivative of f(x) = -(x + 2)^2 + 16 is f′(x) = -2*(x + 2) * 1 + 0, which simplifies to f′(x) = -2x - 4.

Calculus

Find derivatives using the chain rule I

Full solution

f(x) = -(x+2)^2 + 16

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