Identify Function: Identify the function that needs to be differentiated. The function is A(x) = (8+2x)^2.Find Inner Derivative: Find the derivative of the inner function, u(x) = 8+2x. The derivative of u(x) with respect to x is du/dx = 2.Apply Chain Rule: Apply the chain rule. 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. In this case, the derivative of A(x) = (8+2x)^2 is dA/dx = 2*(8+2x)^(2-1)*du/dx.Simplify Expression: Simplify the derivative expression. The derivative of A(x) = (8+2x)^2 is dA/dx = 2*(8+2x)*2.Perform Multiplication: Perform the multiplication to get the final derivative. The derivative of A(x) = (8+2x)^2 is dA/dx = 4*(8+2x).Expand Final Form: Expand the expression to get the final simplified form of the derivative. The derivative of A(x) = (8+2x)^2 is dA/dx = 4*8 + 4*2x = 32 + 8x.

Calculus

Find derivatives using the chain rule I

Full solution

Q. A(x)=(

8

2

x)^

2

Identify Function: Identify the function that needs to be differentiated. The function is $A(x) = (8+2x)^2$ .
Find Inner Derivative: Find the derivative of the inner function, $u(x) = 8+2x$ . The derivative of $u(x)$ with respect to $x$ is $\frac{du}{dx} = 2$ .
Apply Chain Rule: Apply the chain rule. 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. In this case, the derivative of $A(x) = (8+2x)^2$ is $\frac{dA}{dx} = 2\cdot(8+2x)^{2-1}\cdot\frac{du}{dx}$ .
Simplify Expression: Simplify the derivative expression. The derivative of $A(x) = (8+2x)^2$ is $\frac{dA}{dx} = 2\cdot(8+2x)\cdot2$ .
Perform Multiplication: Perform the multiplication to get the final derivative. The derivative of $A(x) = (8+2x)^2$ is $\frac{dA}{dx} = 4\cdot(8+2x)$ .
Expand Final Form: Expand the expression to get the final simplified form of the derivative. The derivative of $A(x) = (8+2x)^2$ is $\frac{dA}{dx} = 4\cdot8 + 4\cdot2x = 32 + 8x$ .