Questions: Given that f(x) = x^9 h(x) h(-1) = 3 h'(−1) = 6 Calculate f'(−1).

Transcript text: Given that \[ \begin{array}{l} f(x)=x^{9} h(x) \\ h(-1)=3 \\ h^{\prime}(-1)=6 \end{array} \] Calculate $f^{\prime}(-1)$.

Solution

Step 1: Apply the product rule for differentiation

Given that $f(x) = x^n h(x)$, we use the product rule $(uv)^{\prime} = u^{\prime}v + uv^{\prime}$.

Step 2: Calculate the derivative of $u = x^n$ using the power rule

Using the power rule, $u^{\prime} = nx^{n-1}$.

Step 3: Substitute $u$, $u^{\prime}$, $v = h(x)$, and $v^{\prime} = h^{\prime}(x)$ into the formula

We substitute $u = x^n$, $u^{\prime} = 9x^{8}$, $v = h(x)$, and $v^{\prime} = h^{\prime}(x)$ into the formula.

Step 4: Evaluate the resulting expression at $x = a$

Substituting $a = -1$, $k = 3$, and $m = 6$ into the formula, we get $f^{\prime}(a) = -1^9 \cdot 6 + 9 \cdot -1^{8} \cdot 3$.

The derivative of the function at $x = -1$ is approximately 21.

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