Differential Calculus Questions 

Differential Calculus is a branch of mathematics that deals with the study of rates of change and slopes of curves. It primarily focuses on the concept of the derivative, which represents how a function changes as its input changes. From analyzing motion to optimizing real-world problems, differential calculus is essential in physics, engineering, economics, and more. It forms the foundation for advanced topics like integration, differential equations, and mathematical modeling.

1.0What is Differential Calculus?

Differential calculus studies how a function changes as its input changes. The central concept is the derivative, which represents the instantaneous rate of change or the slope of the tangent to a curve at a point.

Key Topics Covered in Differential Calculus

• Derivative of a function
• Rules of differentiation
• Chain rule
• Implicit differentiation
• Higher-order derivatives
• Tangents and normals
• Increasing/decreasing functions
• Maxima and minima

2.0Differential Calculus Formulas Recap

3.0Differential Calculus Questions and Answers

Example 1: Find $\frac{d}{dx}(3x^4+2x^2-5x+7)$

Solution:
Using the power rule:

$\frac{d}{dx}(3x^4)=12x^3,\frac{d}{dx}(2x^2)=4x,\frac{d}{dx}(-5x)=-5,\frac{d}{dx}(7)=0$

Final answer:$12x^3+4x-5$

Example 2: Find $\frac{d}{dx}\left(\sin (x^2)\right)$

Solution:
Let $u=x^2$, then:

$\frac{d}{dx}(\sin (x^2))=\cos (x^2)\cdot \frac{d}{dx}(x^2)=\cos (x^2)\cdot 2x=2x\cos (x^2)$

Example 3: Find $\frac{d}{dx}\left(\frac{x^2+1}{x}\right)$

Solution:

Use quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

Here, $u=x^2+1,v=x$

$u^{\prime }=2x,v^{\prime }=1$

$\frac{(2x)(x)-(x^2+1)(1)}{x^2}$

$=\frac{2x^2-x^2-1}{x^2}$

$=\frac{x^2-1}{x^2}$ 

Example 4: Find the slope of the tangent to $y=x^3-3x$ at x = 1

Solution:

$\frac{dy}{dx}=3x^2-3$

At x = 1,

$\frac{dy}{dx}=3(1{)}^2-3=0$

So, the slope of the tangent = 0 (horizontal tangent).

Example 5: Find the maximum and minimum value of $f(x)=x^3-6x^2+9x$

Solution:

First derivative: $f^{\prime }(x)=3x^2-12x+9$

Set f'(x) = 0:

$3x^2-12x+9=0$

$\Rightarrow x^2-4x+3=0$

$\Rightarrow x=1,3$

Second derivative:

f''(x) = 6x - 12 

At x = 1, f''(1) = -6 ⇒ Maxima

At x = 3, f''(3) = 6 ⇒ Minima

4.0Challenging Differential Calculus Questions for Practice

Q1. If $y=ta{n}^{-1}(\sqrt{1-x^2}/x)$, find $\frac{dy}{dx}$

Q2. Find the derivative of $x^x$ using logarithmic differentiation.

Q3. Find the equation of the tangent to the curve $y=ln(x^2+1)$ at x = 1

Q4. Determine the points of inflection for $y=x^4-4x^3+6x^2$

Q5. Find $\frac{dy}{dx}$ if $x^2+y^2=25$ using implicit differentiation.