If
𝑔
(
𝑥
)
+
𝑥
cos
⁡
(
𝑔
(
𝑥
)
)
=
𝑥
9
,
find
𝑔
′
(
0
)
.

Question

If 
𝑔
(
𝑥
)
+
𝑥
cos
⁡
(
𝑔
(
𝑥
)
)
=
𝑥
9
,
 find 
𝑔
′
(
0
)
.

BlackTom AI · Accepted Answer

We start by restating the given equation to anchor the problem: g(x) + x cos(g(x)) = x^9. Our goal is to find g'(0).

Step 1: Determine g(0) from the original equation. Substitute x = 0: g(0) + 0·cos(g(0)) = 0^9, which simplifies to g(0) = 0.

Step 2: Differentiate implicitly with respect to x. Let F(x) = g(x) + x cos(g(x)) − x^9. Since F(x) = 0 for all x, F'(x) = 0 as well. Differentiate term by term:
- The derivative of g(x) is g'(x).
- The derivative of x cos(g(x)) uses the product rule: first derivative of x is 1 times cos(g(x)) plus x times the derivative of cos(g(x)), which is −sin(g(x))·g'(x).
- The derivative of −x^9 is −9x^8.
Putting these together, F'(x) = g'(x) + cos(g(x)) + x[−sin(g(x))·g'(x)] − 9x^8.

Step 3: Evaluate F'(x) at x = 0. We already found g(0) = 0, so cos(g(0)) = cos(0) = 1 and sin(g(0)) = sin(0) = 0. Also the term involving x is zero because x = 0. Thus:
F'(0) = g'(0) + 1 + 0 − 0 = g'(0) + 1.

Step 4: Use the fact that F'(0) = 0 to solve for g'(0). Setting g'(0) + 1 = 0 gives g'(0) = −1.

Summary: By differentiating implicitly and evaluating at x = 0, using g(0) = 0, we arrive at the relation g'(0) + 1 = 0, which yields the value of the derivative at 0. The reasoning steps rely on implicit differentiation, the chain rule for cos(g(x)), and the evaluation of trig functions at the known point g(0) = 0.

If 𝑔 ( 𝑥 ) + 𝑥 cos ⁡ ( 𝑔 ( 𝑥 ) ) = 𝑥 9 , find 𝑔 ′ ( 0 ) .

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