Klein Welch
01/12/2023 · Middle School
If \( f(x)=\sin ^{4} x \), find \( f^{\prime}(x) \) Find \( f^{\prime}(1) \)
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Tutor-Verified Answer
Step-by-step Solution
To find the derivative \( f'(x) \) of the function \( f(x) = \sin^4 x \), we can use the chain rule.
Let \( u = \sin x \), then \( f(x) = u^4 \). The derivative of \( u^4 \) with respect to \( x \) is given by:
\[
f'(x) = \frac{d}{dx}(u^4) = 4u^3 \frac{du}{dx}
\]
Now, we need to find \( \frac{du}{dx} \):
\[
\frac{du}{dx} = \cos x
\]
Substituting back into the derivative, we have:
\[
f'(x) = 4(\sin x)^3 \cos x
\]
Thus, the derivative of \( f(x) \) is:
\[
f'(x) = 4 \sin^3 x \cos x
\]
Next, we need to evaluate \( f'(1) \). We first calculate \( \sin(1) \) and \( \cos(1) \):
\[
f'(1) = 4 \sin^3(1) \cos(1)
\]
Using a calculator or a table, we find:
\[
\sin(1) \approx 0.8415 \quad \text{and} \quad \cos(1) \approx 0.5403
\]
Now we can compute \( f'(1) \):
\[
f'(1) = 4 (0.8415)^3 (0.5403)
\]
Calculating \( (0.8415)^3 \):
\[
(0.8415)^3 \approx 0.5921
\]
Now substituting this value back into the expression for \( f'(1) \):
\[
f'(1) \approx 4 \times 0.5921 \times 0.5403
\]
Calculating this gives:
\[
f'(1) \approx 4 \times 0.5921 \times 0.5403 \approx 1.2801
\]
Thus, the final result is:
\[
\boxed{1.2801}
\]
Quick Answer
\( f'(1) \approx 1.2801 \)
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