Math / Calculus

QuestionLet $f$ be the function defined by $f(x)=\sin (h(x))$ , where $h$ is a differentiable function. Which of the following is equivalent to the derivative of $f$ with respect to $x$ ? (A) $\cos (h(x))$ (B) $\cos (h \prime(x))$ (C) $\cos (h(x)) h \prime(x)$ (D) $\sin (h(x)) h \prime(x)$

Studdy Solution

STEP 1

1. The function $f(x) = \sin(h(x))$ is a composition of functions, specifically the sine function and the function $h(x)$ .
2. The derivative of a composition of functions can be found using the chain rule.
3. $h(x)$ is differentiable, meaning $h'(x)$ exists.

STEP 2

1. Identify the outer and inner functions for applying the chain rule.
2. Apply the chain rule to find the derivative of $f(x)$ .
3. Simplify the derivative expression.
4. Match the simplified expression with the given options.

STEP 3

Identify the outer and inner functions: - Outer function: $\sin(u)$ where $u = h(x)$ - Inner function: $u = h(x)$

STEP 4

Apply the chain rule to find the derivative of $f(x) = \sin(h(x))$ :
The chain rule states that if a function $y = g(u)$ and $u = h(x)$ , then the derivative $\frac{dy}{dx} = g'(u) \cdot h'(x)$ .
For $f(x) = \sin(h(x))$ , we have: $f'(x) = \cos(h(x)) \cdot h'(x)$

STEP 5

Simplify the expression for the derivative:
The expression $\cos(h(x)) \cdot h'(x)$ is already simplified.

STEP 6

Match the simplified expression with the given options:
The expression $\cos(h(x)) \cdot h'(x)$ matches option (C).
The derivative of $f(x)$ with respect to $x$ is equivalent to option (C): $\cos(h(x)) h'(x)$ .

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QuestionLet $f$ be the function defined by $f(x)=\sin (h(x))$ , where $h$ is a differentiable function. Which of the following is equivalent to the derivative of $f$ with respect to $x$ ? (A) $\cos (h(x))$ (B) $\cos (h \prime(x))$ (C) $\cos (h(x)) h \prime(x)$ (D) $\sin (h(x)) h \prime(x)$

Studdy Solution

STEP 1

1. The function $f(x) = \sin(h(x))$ is a composition of functions, specifically the sine function and the function $h(x)$ .
2. The derivative of a composition of functions can be found using the chain rule.
3. $h(x)$ is differentiable, meaning $h'(x)$ exists.

STEP 2

1. Identify the outer and inner functions for applying the chain rule.
2. Apply the chain rule to find the derivative of $f(x)$ .
3. Simplify the derivative expression.
4. Match the simplified expression with the given options.

STEP 3

Identify the outer and inner functions: - Outer function: $\sin(u)$ where $u = h(x)$ - Inner function: $u = h(x)$

STEP 4

Apply the chain rule to find the derivative of $f(x) = \sin(h(x))$ :
The chain rule states that if a function $y = g(u)$ and $u = h(x)$ , then the derivative $\frac{dy}{dx} = g'(u) \cdot h'(x)$ .
For $f(x) = \sin(h(x))$ , we have: $f'(x) = \cos(h(x)) \cdot h'(x)$

STEP 5

Simplify the expression for the derivative:
The expression $\cos(h(x)) \cdot h'(x)$ is already simplified.

STEP 6

Match the simplified expression with the given options:
The expression $\cos(h(x)) \cdot h'(x)$ matches option (C).
The derivative of $f(x)$ with respect to $x$ is equivalent to option (C): $\cos(h(x)) h'(x)$ .

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Studdy Solution

STEP 1

STEP 2

1. Identify the outer and inner functions for applying the chain rule.2. Apply the chain rule to find the derivative of f(x) f(x) f(x).3. Simplify the derivative expression.4. Match the simplified expression with the given options.

STEP 3

Identify the outer and inner functions: - Outer function: sin⁡(u) \sin(u) sin(u) where u=h(x) u = h(x) u=h(x) - Inner function: u=h(x) u = h(x) u=h(x)

STEP 4

STEP 5

Simplify the expression for the derivative:The expression cos⁡(h(x))⋅h′(x) \cos(h(x)) \cdot h'(x) cos(h(x))⋅h′(x) is already simplified.

STEP 6

Match the simplified expression with the given options:The expression cos⁡(h(x))⋅h′(x) \cos(h(x)) \cdot h'(x) cos(h(x))⋅h′(x) matches option (C). The derivative of f(x) f(x) f(x) with respect to x x x is equivalent to option (C): cos⁡(h(x))h′(x) \cos(h(x)) h'(x) cos(h(x))h′(x).

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1. Identify the outer and inner functions for applying the chain rule.
2. Apply the chain rule to find the derivative of $f(x)$ .
3. Simplify the derivative expression.
4. Match the simplified expression with the given options.

Identify the outer and inner functions: - Outer function: $\sin(u)$ where $u = h(x)$ - Inner function: $u = h(x)$

Simplify the expression for the derivative:
The expression $\cos(h(x)) \cdot h'(x)$ is already simplified.

Match the simplified expression with the given options:
The expression $\cos(h(x)) \cdot h'(x)$ matches option (C).
The derivative of $f(x)$ with respect to $x$ is equivalent to option (C): $\cos(h(x)) h'(x)$ .