Question

If \(y=\left(x^{2}+7 x+2\right)\left(e^{x}-\log x\right)\) and \((d y / d x)\) \(=\left(x^{2}+A x+B\right)\left|e^{x}-\\{1 / x\\}\right|+\left(e^{x}-\log x\right)(C x+D)\) then \(\mathrm{A}+\mathrm{B}-\mathrm{C}-\mathrm{D}=\) (a) 0 (b) 7 (c) 2 (d) 9

Short answer

The value of \(A + B - C - D\) is 0. The correct answer is (a) 0.

Step-by-step solution

Identify the functions u and v

Let's define \(u(x) = x^2 + 7x + 2\) and \(v(x) = e^x - \log x\). Then, \(y(x) = u(x)v(x)\).

Calculate the derivatives of u and v

We find the derivatives of u and v with respect to x: \(\frac{du}{dx} = 2x + 7\) \(\frac{dv}{dx} = e^x - \frac{1}{x}\)

Apply the product rule

Now, let's apply the product rule to find the derivative of y with respect to x: \(\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx} = (x^2 + 7x + 2) (e^x - \frac{1}{x}) + (e^x - \log x)(2x + 7)\)

Compare the obtained expression with the given one

Now we have the expression for \(\frac{dy}{dx}\) in the form: \((x^2 + 7x + 2) \left(e^x - \frac{1}{x}\right) + (e^x - \log x)(2x + 7)\) Comparing it to the given expression: \((x^2 + Ax + B) \left| e^x - \frac{1}{x}\right| + (e^x - \log x)(Cx + D)\) We can identify the constants A, B, C, and D as follows: \(A = 7\) \(B = 2\) \(C = 2\) \(D = 7\)

Calculate the sum A + B - C - D

Now, let's calculate the sum \(A + B - C - D\): \(A + B - C - D = (7) + (2) - (2) - (7) = 0\) So, the value of \(A + B - C - D = 0\). The correct answer is (a) 0.

Key concepts

Understanding the Product Rule

When you want to find the derivative of a product of two functions, you'll need the **product rule**. This rule is essential when dealing with expressions like \(y = u(x) \cdot v(x)\). If \(u(x)\) and \(v(x)\) are two differentiable functions, then the derivative of their product \(y\) is given by:

• \(\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}\)

This formula essentially helps you break down the derivative into simpler parts. You differentiate each function separately and sum the results. You multiply the first function by the derivative of the second and then add the second function multiplied by the derivative of the first.
This technique maintains the relationship between the two functions while allowing you to find the overall rate of change.

What are Derivatives?

**Derivatives** represent the rate of change of a function concerning a variable. If you think of a graph, a derivative gives you the slope of the tangent line to the curve at any given point.

• For simple functions, like \(f(x) = x^2\), the derivative tells you how fast \(x^2\) is changing at any value of \(x\).
• The derivative of \(f(x) = x^2\) is \(f'(x) = 2x\).

Derivatives help you understand physical and abstract changes in real-world scenarios. They're used in physics to find velocity from position, in economics to find marginal costs, and in statistics to model trends.
When working with derivatives, remember they require that the function is smooth and continuous over the interval you're examining.

Functions and Their Role

**Functions** are mathematical entities that relate an input to an output. They're represented as \(f(x)\), which assigns each input \(x\) to an output \(f(x)\).

• In our case, the functions \(u(x) = x^2 + 7x + 2\) and \(v(x) = e^x - \log x\) represent how "complex" inputs change and interact.
• Functions can model real-world phenomena, such as financial growth or the temperature of a location over time.

Functions are the backbone of calculus; they allow us to study change and motion in a structured way. When two functions are combined in multiplication, differentiation becomes more complex, necessitating methods like the product rule.
Understanding functions is essential for any calculus-related topic.

Introduction to Integration

While our focus here is on differentiation, it's useful to note the connection to **integration**. Integration is like the reverse operation of differentiation.

• When you integrate a function, you're finding the area under its curve.
• If \(F(x)\) is the integral of \(f(x)\), then \(\frac{d}{dx}F(x) = f(x)\).

Integration helps in calculating quantities like total distance traveled given a speed function or total income given a rate of income over time.
This dual relationship between differentiation and integration forms the foundation of calculus, allowing deeper exploration into continuous change and accumulation.