Derivatives

Many types of AP calculus problems require knolwdge of derivatives of functions.

1. The derivative of a differentiable function needs to be determined frequently in calculus problems with minimum, maximum or the inflexion points of a function. Examples.

2. The product rule can be applied to determine the derivatives of some functions: If two functions f(x) and g(x) are differentiable, then the product is differentiable and (fg)' = f 'g + g' f. Examples.

3. The quotient rule can be applied to determine the derivatives of some functions: If two functions f(x) and g(x) are differentiable, then the quotient is differentiable and (f/g)' = (f 'g - g' f)/g². Examples.

4. The chain rule can be applied to determine the derivatives of some functions: If two functions f(x) and g(x) are differentiable, y = f(u) and u = g(x), then the derivative of y is:
dy/dx = (dy/du)(du/dx) Examples.

Examples

Question 1:

The volume of water in a lake is given by the function V(t) = 2·10⁶ + 8·10⁶e^t - 3·10⁵e^2t, where t is in years and the volume is in cubic meters.
Determine the time t > 0 when the volume of water is maximum in the lake.

Solution:
V'(t) = 8·10⁶e^t - 6·10⁵e^2t
V'(t) = 0
8·10⁶e^t - 6·10⁵e^2t = 0
4·10⁶e^t - 3·10⁵e^2t = 0
e^t(4·10⁶ - 3·10⁵e^t) = 0
e^t = 40/3
t = ln(40/3)
t = 2.59 years

Question 2:

Calculate the derivative of the function f(x) = e^x(x² + 1).

Solution:
If we consider g(x) = e^x and h(x) = x² + 1, then f(x) = g(x)h(x).
g '(x) = e^x
h'(x) = 2x
f'(x) = e^x(x² + 1) + e^x2x = e^x(x² + 2x + 1) = e^x(x + 1)².

Question 3:

Calculate the derivative of the function:
ab calculus, limits, question, calculation

Solution:

ab calculus, derivative, question, calculation

Question 4:

Calculate the derivative of the function f(x) = cos(x³ + x² + 1).

Solution:
u = x³ + x² + 1
du/dx = 3x² + 2x
df/du = -sin(u)
df/dx = (df/du)(du/dx) = -(3x² + 2x)sin(x³ + x² + 1)

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