Problem 6.3 Calculate The Following Derivative

Introduction & Importance

Calculating the derivative of a function is a fundamental concept in calculus. It measures how a function changes as its input changes. Understanding derivatives is crucial for various applications, such as physics, engineering, economics, and data analysis.

How to Use This Calculator

1. Enter the function for which you want to find the derivative.
2. Enter the variable with respect to which you want to differentiate.
3. Click the “Calculate” button.

Formula & Methodology

The derivative of a function f(x) with respect to x is given by:

f'(x) = lim (h→0) [f(x+h) – f(x)] / h

Or, using the difference quotient:

f'(x) = [f(x + Δx) – f(x)] / Δx

Real-World Examples

Example 1: Velocity

If the position of an object is given by s(t) = 3t^2 – 4t + 1, find its velocity v(t).

v(t) = s'(t) = 6t – 4

Example 2: Revenue

If the revenue R(p) of a company is given by R(p) = 1000p – 0.1p^2, find the marginal revenue MR(p).

MR(p) = R'(p) = 1000 – 0.2p

Example 3: Profit

If the profit π(q) of a firm is given by π(q) = 20q – 0.01q^2 – 500, find the marginal profit MP(q).

MP(q) = π'(q) = 20 – 0.02q

Data & Statistics

Derivatives of common functions

Function	Derivative
x^n	n*x^(n-1)
sin(x)	cos(x)
cos(x)	-sin(x)

Derivatives of some elementary functions

Function	Derivative
e^x	e^x
ln(x)	1/x
a^x	a^x * ln(a)

Expert Tips

• Use the power rule, product rule, quotient rule, and chain rule to differentiate complex functions.
• Implicit differentiation can be used to differentiate implicitly defined functions.
• Always check your answers by plugging them back into the original function.

Interactive FAQ

What is the difference between a derivative and a difference quotient?

The difference quotient is an approximation of the derivative. The derivative is the exact rate of change of a function at a specific point, while the difference quotient is an approximation of that rate of change.

How do I find the derivative of a function at a specific point?

To find the derivative of a function at a specific point, substitute the value of the variable into the derivative of the function and evaluate.

What is the slope of the tangent line to a curve at a specific point?

The slope of the tangent line to a curve at a specific point is given by the derivative of the function at that point.

For more information, see the Khan Academy guide on differentiation.