consider the function (f(x) = (8 - 7x)^6).

step 3 of 3: find an equation for the tangent line to the graph of (f(x) = (8 - 7x)^6) at the point ((1, 1)).

answer 2 points
(y =)

Question

consider the function (f(x) = (8 - 7x)^6).

step 3 of 3: find an equation for the tangent line to the graph of (f(x) = (8 - 7x)^6) at the point ((1, 1)).

answer 2 points
(y =)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Chain Rule Differentiation", "Tangent Line Equation" ], "new_concepts": [], "current_concepts": [ "Chain Rule Differentiation", "Tangent Line Equation" ] } ## Differentiate the function Using the *Chain Rule Differentiation* knowledge point $$ \begin{aligned} f(x) &= (8 - 7x)^6 \ f'(x) &= 6(8 - 7x)^5 \cdot \frac{d}{dx}(8 - 7x) \ f'(x) &= 6(8 - 7x)^5 \cdot (-7) \ f'(x) &= -42(8 - 7x)^5 \end{aligned} $$ ## Calculate the slope at the given point Using the *Chain Rule Differentiation* knowledge point $$ \begin{aligned} m &= f'(1) \ m &= -42(8 - 7(1))^5 \ m &= -42(1)^5 \ m &= -42 \end{aligned} $$ ## Find the tangent line equation Using the *Tangent Line Equation* knowledge point $$ \begin{aligned} y - y_1 &= m(x - x_1) \ y - 1 &= -42(x - 1) \ y - 1 &= -42x + 42 \ y &= -42x + 43 \end{aligned} $$ Find an equation for the tangent line to the graph of $f(x) = (8 - 7x)^6$ at the point $(1, 1)$. $y =$ $-42x + 43$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Tangent Line Equation" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Differentiate the function

Calculate the slope at the given point

Find the tangent line equation

Answer: