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A(x)=( $8$ + $2$ x)^ $2$

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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 $\frac{dA}{dx} = 4\cdot8 + 4\cdot2x = 32 + 8x$ .

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