What is the derivative of sin cubed?
Understand the Problem
The question is asking for the derivative of the function sin^3(x). This involves applying the chain rule of differentiation, where we first differentiate the outer function (cubing) and then multiply it by the derivative of the inner function (sin(x)).
Answer
The derivative is given by $\frac{dy}{dx} = 3\sin^2(x) \cos(x)$.
Answer for screen readers
The derivative of the function $y = \sin^3(x)$ is given by: $$ \frac{dy}{dx} = 3\sin^2(x) \cos(x) $$
Steps to Solve
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Identify the function and apply the chain rule We need to differentiate the function $y = \sin^3(x)$. We can express this as $y = (\sin(x))^3$. To differentiate, we will apply the chain rule, which states that if we have a function of another function, we need to differentiate the outer function and multiply it by the derivative of the inner function.
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Differentiate the outer function The outer function here is $u^3$ where $u = \sin(x)$. The derivative of $u^3$ with respect to $u$ is $3u^2$. Therefore, we differentiate the outer function as follows: $$ \frac{dy}{du} = 3(\sin(x))^2 $$
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Differentiate the inner function Next, we need to find the derivative of the inner function $u = \sin(x)$. The derivative of $\sin(x)$ with respect to $x$ is: $$ \frac{du}{dx} = \cos(x) $$
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Applying the chain rule According to the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
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Substituting the derivatives we found Now we can substitute what we calculated in the previous steps: $$ \frac{dy}{dx} = 3(\sin(x))^2 \cdot \cos(x) $$
The derivative of the function $y = \sin^3(x)$ is given by: $$ \frac{dy}{dx} = 3\sin^2(x) \cos(x) $$
More Information
This derivative represents the rate of change of the function $\sin^3(x)$ with respect to $x$. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions.
Tips
- Forgetting to apply the chain rule: Often, students might neglect to differentiate the inner function when applying the chain rule. To avoid this, always remember to consider both the outer and inner functions.
- Confusion between the derivative of sine and cosine: Make sure to remember that the derivative of $\sin(x)$ is $\cos(x)$, and vice versa.
