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  1. AP Calculus
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How do you find the derivative of a polynomial?

Apply the power rule to each term: multiply by the exponent and reduce the exponent by 1. Use the sum/difference rule for multiple terms.

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How do you find the derivative of a polynomial?

Apply the power rule to each term: multiply by the exponent and reduce the exponent by 1. Use the sum/difference rule for multiple terms.

How do you find the derivative of a product of two functions?

Use the product rule: (first function) * (derivative of second) + (second function) * (derivative of first).

How do you find the derivative of a quotient of two functions?

Use the quotient rule: [(bottom) * (derivative of top) - (top) * (derivative of bottom)] / (bottom)^2.

How do you find the derivative of f(x)=ekxf(x) = e^{kx}f(x)=ekx?

Use the chain rule: f′(x)=kekxf'(x) = ke^{kx}f′(x)=kekx.

How do you find the derivative of f(x)=ln⁡(g(x))f(x) = \ln(g(x))f(x)=ln(g(x))?

Use the chain rule: f′(x)=g′(x)g(x)f'(x) = \frac{g'(x)}{g(x)}f′(x)=g(x)g′(x)​.

How do you find the equation of a tangent line to f(x)f(x)f(x) at x=ax=ax=a?

  1. Find f(a)f(a)f(a). 2. Find f′(x)f'(x)f′(x). 3. Find f′(a)f'(a)f′(a) (the slope). 4. Use point-slope form: y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)y−f(a)=f′(a)(x−a).

How do you find where a function is increasing or decreasing?

  1. Find f′(x)f'(x)f′(x). 2. Find critical points where f′(x)=0f'(x) = 0f′(x)=0 or is undefined. 3. Test intervals around critical points to determine the sign of f′(x)f'(x)f′(x).

How do you find local maxima and minima?

  1. Find critical points. 2. Use the first derivative test (sign change of f′(x)f'(x)f′(x)) or the second derivative test (sign of f′′(x)f''(x)f′′(x)).

How do you find intervals of concavity?

  1. Find f′′(x)f''(x)f′′(x). 2. Find points where f′′(x)=0f''(x) = 0f′′(x)=0 or is undefined. 3. Test intervals around these points to determine the sign of f′′(x)f''(x)f′′(x).

How do you find points of inflection?

  1. Find f′′(x)f''(x)f′′(x). 2. Find points where f′′(x)=0f''(x) = 0f′′(x)=0 or is undefined. 3. Check that the concavity changes at these points.

What is the power rule?

If f(x)=xnf(x) = x^nf(x)=xn, then f′(x)=nxn−1f'(x) = nx^{n-1}f′(x)=nxn−1.

What is the product rule?

If h(x)=f(x)g(x)h(x) = f(x)g(x)h(x)=f(x)g(x), then h′(x)=f′(x)g(x)+f(x)g′(x)h'(x) = f'(x)g(x) + f(x)g'(x)h′(x)=f′(x)g(x)+f(x)g′(x).

What is the quotient rule?

If h(x)=f(x)g(x)h(x) = \frac{f(x)}{g(x)}h(x)=g(x)f(x)​, then h′(x)=f′(x)g(x)−f(x)g′(x)(g(x))2h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}h′(x)=(g(x))2f′(x)g(x)−f(x)g′(x)​.

What is the derivative of sin(x)?

ddxsin⁡(x)=cos⁡(x)\frac{d}{dx} \sin(x) = \cos(x)dxd​sin(x)=cos(x)

What is the derivative of cos(x)?

ddxcos⁡(x)=−sin⁡(x)\frac{d}{dx} \cos(x) = -\sin(x)dxd​cos(x)=−sin(x)

What is the derivative of tan(x)?

ddxtan⁡(x)=sec⁡2(x)\frac{d}{dx} \tan(x) = \sec^2(x)dxd​tan(x)=sec2(x)

What is the derivative of exe^xex?

ddxex=ex\frac{d}{dx} e^x = e^xdxd​ex=ex

What is the derivative of ln(x)?

ddxln⁡(x)=1x\frac{d}{dx} \ln(x) = \frac{1}{x}dxd​ln(x)=x1​

What is the constant rule?

If f(x)=cf(x) = cf(x)=c, where c is a constant, then f′(x)=0f'(x) = 0f′(x)=0.

What is the constant multiple rule?

ddx[cf(x)]=c⋅f′(x)\frac{d}{dx}[cf(x)] = c \cdot f'(x)dxd​[cf(x)]=c⋅f′(x)

What is the sum rule?

ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)dxd​[f(x)+g(x)]=f′(x)+g′(x)

Explain the concept of instantaneous rate of change.

The rate at which a function's output changes with respect to its input at a specific point. It's the derivative.

Explain when a derivative does not exist.

Derivatives don't exist at sharp corners, cusps, vertical tangents, or points of discontinuity.

What does the sign of the derivative tell you?

A positive derivative indicates the function is increasing, a negative derivative indicates it is decreasing, and a zero derivative indicates a stationary point.

Explain the relationship between differentiability and continuity.

If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.

What is the geometric interpretation of the derivative?

The derivative at a point represents the slope of the tangent line to the curve at that point.

Explain how to find the equation of a tangent line.

Find the derivative at the given point to get the slope. Use the point-slope form of a line: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).

How can derivatives be used in optimization problems?

Derivatives can be used to find maximum and minimum values of a function by finding critical points (where the derivative is zero or undefined).

Explain the concept of higher-order derivatives.

Higher-order derivatives are derivatives of derivatives. For example, the second derivative is the derivative of the first derivative, and so on.

What does the second derivative tell you about a function?

The second derivative tells you about the concavity of a function. A positive second derivative indicates concave up, and a negative second derivative indicates concave down.

Explain the difference between average and instantaneous rate of change.

Average rate of change is over an interval, while instantaneous rate of change is at a single point.