Question text 3Marks Differentiate the following function at x=1 using the chain rule: [math: f(x)=(4x3−4x2+6x−7)5]. The derivative is [math: f′(1)=] Answer 1[input].Notes Report question issue Question 5 Notes

Question

BlackTom AI · Accepted Answer

Chain rule: if f(x) = [g(x)]^5 then f′(x) = 5[g(x)]^4 · g′(x).

1) Relevant knowledge points and concepts
- Chain rule for a composite function (outer power function and inner polynomial).
- Power rule: d/dx[u^5] = 5u^4 · u′.
- Polynomial differentiation term-by-term.

2) Complete reasoning step by step
Let g(x) = 4x^3 − 4x^2 + 6x − 7. Then f(x) = [g(x)]^5. By the chain rule,
f′(x) = 5[g(x)]^4 · g′(x).
Compute g′(x) by differentiating term-by-term:
g′(x) = d/dx(4x^3) − d/dx(4x^2) + d/dx(6x) − d/dx(7)
       = 12x^2 − 8x + 6.

3) Calculations evaluated at x = 1 (all intermediate steps)
First evaluate the inner function at x = 1:
g(1) = 4(1)^3 − 4(1)^2 + 6(1) − 7 = 4 − 4 + 6 − 7 = 0.
(With g(1) = 0, the fourth power is g(1)^4 = 0^4 = 0.)
Next evaluate the derivative of the inner at x = 1:
g′(1) = 12(1)^2 − 8(1) + 6 = 12 − 8 + 6 = 10.
Now combine in the chain rule formula:
f′(1) = 5 · g(1)^4 · g′(1) = 5 · 0 · 10 = 0.

4) Why the correct answer is correct
Because the outer factor includes g(1)^4, and g(1) = 0, the whole product becomes zero regardless of the nonzero value of g′(1). Any time the inner function equals zero at the point, the derivative of [g(x)]^5 at that point is zero due to the factor g(1)^4.

5) Brief note on other possible results
A straightforward but different numeric evaluation of g(1) would lead to a nonzero result (for example, if g(1) were −1 then f′(1) would be 5·1·g′(1) = 50). However, because g(1) = 0 here, those alternatives are not applicable.

6) Conclusion confirming the final answer
Following the chain rule and the evaluations above, f′(1) = 0, which matches the provided answer.

Question text 3Marks Differentiate the following function at x=1 using the chain rule: [math: f(x)=(4x3−4x2+6x−7)5]. The derivative is [math: f′(1)=] Answer 1[input].Notes Report question issue Question 5 Notes多项填空题

类似问题

Which of these is the result when you differentiate the expression ln ⁡ ( 3 cos ⁡ ( 𝑥 ) ) with respect to 𝑥 ?

Differentiate 5 sin ⁡ ( 𝑥 3 + 2 𝑥 )

Which rule would be used to differentiate the function

Find dy/dt. y = cos()

If wanting to differentiate: 𝑦 = 1 𝑥 2 + 1 The most appropriate rule to use would be the [ Select ] chain general right handed product rule

If f ( x ) = 5 x − 1 , g ( 0 ) = 2 , g ′ ( 0 ) = 4 , and h ( x ) = f ( g ( x ) ) {"version":"1.1","math":"f(x)=5x-1,\ g(0)=2,\ g'(0)=4, \text{ and } h(x)=f(g(x))"}then what is the value of h ′ ( 0 ) ? {"version":"1.1","math":"h'(0)?"}

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单