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How do you differentiate $y = \sin(x^2)$ using the chain rule?

Identify inner ( $u = x^2$ ) and outer ( $y = \sin(u)$ ) functions. 2. Differentiate: $\frac{dy}{du} = \cos(u)$ , $\frac{du}{dx} = 2x$ . 3. Apply chain rule: $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$ .

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How do you differentiate $y = \sin(x^2)$ using the chain rule?

Identify inner ( $u = x^2$ ) and outer ( $y = \sin(u)$ ) functions. 2. Differentiate: $\frac{dy}{du} = \cos(u)$ , $\frac{du}{dx} = 2x$ . 3. Apply chain rule: $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$ .

How do you find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$ using implicit differentiation?

Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$ . 2. Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = -\frac{x}{y}$ .

Steps to find higher-order derivatives?

Find the first derivative, $f'(x)$ . 2. Find the derivative of $f'(x)$ to get the second derivative, $f''(x)$ . 3. Repeat to find higher derivatives.

Explain the chain rule.

Differentiate the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

Explain implicit differentiation.

Differentiate both sides of the equation with respect to $x$ , applying the chain rule to terms involving $y$ , and solve for $\frac{dy}{dx}$ .

What does the second derivative tell you?

The concavity of the original function and helps find points of inflection.

What does the first derivative tell you?

The slope of the original function.

How do you find the derivative from a graph?

The derivative at a point is the slope of the tangent line at that point.

What does a positive first derivative on a graph indicate?

The function is increasing.

What does a negative first derivative on a graph indicate?

The function is decreasing.

What does a positive second derivative on a graph indicate?

The function is concave up.

What does a negative second derivative on a graph indicate?

The function is concave down.

What does a point of inflection on a graph indicate?

The concavity of the function changes at that point; $f''(x) = 0$ or is undefined.

How do you find critical points on a graph?

Look for points where the slope (derivative) is zero or undefined.

How do you determine intervals of increasing/decreasing from a graph?

Identify where the slope (derivative) is positive (increasing) or negative (decreasing).

All Flashcards

How do you differentiate $y = \sin(x^2)$ using the chain rule?

Identify inner ( $u = x^2$ ) and outer ( $y = \sin(u)$ ) functions. 2. Differentiate: $\frac{dy}{du} = \cos(u)$ , $\frac{du}{dx} = 2x$ . 3. Apply chain rule: $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$ .

How do you find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$ using implicit differentiation?

Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$ . 2. Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = -\frac{x}{y}$ .

Steps to find higher-order derivatives?

Find the first derivative, $f'(x)$ . 2. Find the derivative of $f'(x)$ to get the second derivative, $f''(x)$ . 3. Repeat to find higher derivatives.

Explain the chain rule.

Differentiate the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

Explain implicit differentiation.

Differentiate both sides of the equation with respect to $x$ , applying the chain rule to terms involving $y$ , and solve for $\frac{dy}{dx}$ .

What does the second derivative tell you?

The concavity of the original function and helps find points of inflection.

What does the first derivative tell you?

The slope of the original function.

How do you find the derivative from a graph?

The derivative at a point is the slope of the tangent line at that point.

What does a positive first derivative on a graph indicate?

The function is increasing.

What does a negative first derivative on a graph indicate?

The function is decreasing.

What does a positive second derivative on a graph indicate?

The function is concave up.

What does a negative second derivative on a graph indicate?

The function is concave down.

What does a point of inflection on a graph indicate?

The concavity of the function changes at that point; $f''(x) = 0$ or is undefined.

How do you find critical points on a graph?

Look for points where the slope (derivative) is zero or undefined.

How do you determine intervals of increasing/decreasing from a graph?

Identify where the slope (derivative) is positive (increasing) or negative (decreasing).