Question 2 (1 point) Listen Find the derivative of the function.
The Correct Answer and Explanation is:
Answer
f′(x)=7(2x4−5x+1)6(8x3−5)f′(x)=7(2x4−5x+1)6(8x3−5)Explanation
The given function,
f(x)=(2x4−5x+1)7f(x)=(2x4−5x+1)7, is a composite function. To find its derivative, the Chain Rule must be applied. The Chain Rule is the fundamental method for differentiating a function nested inside another function. It states that the derivative of a composite function
h(x)=g(u(x))h(x)=g(u(x))is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In formal notation, if
f(x)=g(u(x))f(x)=g(u(x)), then
f′(x)=g′(u(x))⋅u′(x)f′(x)=g′(u(x))⋅u′(x).
First, identify the outer and inner functions.
The outer function is the power of 7, which can be represented as
g(u)=u7g(u)=u7.
The inner function is the polynomial inside the parentheses,
u(x)=2x4−5x+1u(x)=2x4−5x+1.
Next, find the derivative of each of these functions separately.
The derivative of the outer function,
g(u)=u7g(u)=u7, is found using the Power Rule. The derivative is
g′(u)=7u6g′(u)=7u6.
The derivative of the inner function,
u(x)=2x4−5x+1u(x)=2x4−5x+1, is found by applying the Power Rule to each term. The derivative is
u′(x)=2(4x3)−5(1)+0u′(x)=2(4x3)−5(1)+0, which simplifies to
u′(x)=8x3−5u′(x)=8x3−5.
Finally, assemble these components according to the Chain Rule formula,
f′(x)=g′(u(x))⋅u′(x)f′(x)=g′(u(x))⋅u′(x). Substitute the expression for the inner function,
u(x)u(x), back into the derivative of the outer function,
g′(u)g′(u), and multiply by the derivative of the inner function,
u′(x)u′(x).
This yields:
f′(x)=7(2x4−5x+1)6⋅(8x3−5)f′(x)=7(2x4−5x+1)6⋅(8x3−5)This expression is the final derivative of the original function. It is conventional to write the polynomial factor after the composite term.
