Derivative Function Calculus

In mathematics calculus, the derivative of a function at a particular point provides the slope of the tangent line to the function's graph at that point. This slope represents the rate of change of the function with respect to its input variable. If we denote a function by f(x), its derivative is often represented as $f^{\prime }(x)\text{ or }\frac{df}{dx}$

1.0Introduction to Function Derivatives

What is a Derivative?

At its simplest, the derivative of a function at a given point is the slope of the tangent line to the function's graph at that point. This slope indicates how steeply the function's value is changing at that specific input value. Mathematically, for a function f(x), the derivative at x = a is defined as:

$f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

This limit, if it exists, captures the instantaneous rate of change of the function.

2.0Derivative of a Function

The derivative of a function f(x) at x = a is defined as the limit of the function's average rate of change as the interval around a approaches zero. Mathematically, this is expressed as:

$f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

If this limit exists, f'(a) represents the instantaneous rate of change of the function at x = a. The process of finding the derivative is called differentiation.

Notation

Several notations are commonly used to denote the derivative of a function:

• f'(x) (Leibniz notation)
• $\frac{\mathrm{df}}{\mathrm{dx}}$ (Leibniz notation)
• Df(x) (Euler notation)

For higher-order derivatives, which represent the derivative of the derivative, the notation extends to f''(x) for the second derivative, f'''(x) for the third derivative, and so on.

3.0Derivative Function Calculus Rules

Basic Derivative Function Formulas are

1. Constant Function

$\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{c}]=0Wherecisaconstant.$

2. Power Rule

$\frac{d}{dx}\left[x^n\right]=n{x}^{n-1}$ Applicable for any real number n.

3. Constant Multiple Rule

$\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{c}\times \mathrm{f}(\mathrm{x})]=\mathrm{c}\times {\mathrm{f}}^{\prime }(\mathrm{x})$, where c is a constant.

4. Sum Rule

$\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})]={\mathrm{f}}^{\prime }(\mathrm{x})+{\mathrm{g}}^{\prime }(\mathrm{x})$

5. Difference Rule

$\frac{d}{dx}[f(x)-g(x)]=f^{\prime }(x)-g^{\prime }(x)$

6. Product Rule

$\frac{d}{dx}[f(x)\times g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

If 3 functions are involved 

$\frac{d}{dx}[f(x)\times g(x)\times h(x)]=f^{\prime }(x)g(x)h(x)+f(x)g^{\prime }(x)h(x)+f(x)g(x)h^{\prime }(x)$

7. Quotient Rule

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{g(x{)}^2}$

8. Chain Rule

$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\times g^{\prime }(x)$

4.0Derivatives of Standard Functions

1. Exponential Functions

• $\frac{d}{dx}\left[e^x\right]=e^x$
• $\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{a}}^{\mathrm{x}}\right]={\mathrm{a}}^{\mathrm{x}}\ln (\mathrm{a})$

    Where a is a constant.

2. Logarithmic Functions

• $\frac{d}{dx}[\ln (x)]=\frac{1}{x}$
• $\frac{d}{dx}\left[{\log }_a(x)\right]=\frac{1}{x\ln (a)}$

3. Trigonometric Functions

• $\frac{d}{dx}[\sin (x)]=\cos (x)$
• $\frac{d}{dx}[\cos (x)]=-\sin (x)$
• $\frac{d}{dx}[\tan (x)]={\sec }^2(x)$
• $\frac{d}{dx}[\mathrm{cosec}(x)]=-\mathrm{cosec}(x)\cot (x)$
• $\frac{d}{dx}[\sec (x)]=\sec (x)\tan (x)$
• $\frac{d}{dx}[\cot (x)]=-{\mathrm{cosec}}^2(x)$

4. Inverse Trigonometric Functions

• $\frac{d}{dx}[\arcsin (x)]=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[\arccos (x)]=-\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[\arctan (x)]=\frac{1}{1+x^2}$
• $\frac{d}{dx}[\mathrm{arccot}(x)]=-\frac{1}{1+x^2}$
• $\frac{d}{dx}[\mathrm{arcsec}(x)]=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}[\mathrm{arccosec}(x)]=-\frac{1}{|x|\sqrt{x^2-1}}$

5. Implicit Differentiation

For functions not explicitly solved for one variable in terms of another, implicit differentiation is used:

• Implicit Differentiation

$\text{ If }F(x,y)=0\text{, then }\frac{dF}{dx}=F_x^{\prime }+F_y^{\prime }\cdot \frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{F_x^{\prime }}{F_y^{\prime }}$

To find $\frac{dy}{dx}$ of implicit curves, we differentiate each term with respect to x regarding y as a function of x and then collect term with \frac{d y}{d x} together on one side

Differentiation of homogeneous relation

$\frac{dy}{dx}=\frac{y}{x}\frac{d^{2}y}{d{x}^2}=0$

5.0Higher-Order Derivatives

1. Second Derivative (f''(x))

$f^{\prime \prime }(x)=\frac{d}{dx}\left[f^{\prime }(x)\right]$

The second derivative represents the rate of change of the first derivative, providing information about the concavity of the function.

2. n^th Derivative (fⁿ(x))

$f^{(n)}(x)=\frac{d^n}{d{x}^n}[f(x)]$

Higher-order derivatives can provide deeper insights into the behavior of functions, such as inflection points and more detailed curvature information.

6.0Derivative Function Calculus Examples

1. Differentiate $f(x)=x^4$

Solution: Using the power rule $\frac{d}{dx}\left[x^n\right]=n{x}^{n-1}$,$f^{\prime }(x)=4x^3$

2. Differentiate $g(x)=x^3-2x^2+x-5$

Solution: Using the sum and difference rule,

${\mathrm{g}}^{\prime }(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^3\right]-\frac{\mathrm{d}}{\mathrm{dx}}\left[2{\mathrm{x}}^2\right]+\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{x}]-\frac{\mathrm{d}}{\mathrm{dx}}[5]$

$g^{\prime }(x)=3x^2-4x+1$

3. Differentiate $h(x)=x^2\sin (x)$

Solution: Using the product rule $\frac{d}{dx}[u(x)v(x)]=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$,

Let u(x) = x² and v(x) = sin(x),

u'(x) = 2x and v'(x) = cos(x)

Therefore,

h'(x) = (2x) sin(x) + (x²) cos(x)

h'(x) = 2xsin(x) + x²cos(x)

4. Differentiate $k(x)=\frac{x^2+1}{x}$

Solution: Using the quotient rule $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{v(x{)}^2}$,

Let u(x) = x² + 1 and v(x) = x,

u'(x) = 2x and v'(x) = 1

Therefore,

$k^{\prime }(x)=\frac{(2x)(x)-\left(x^2+1\right)(1)}{x^2}{\mathrm{k}}^{\prime }(\mathrm{x})=\frac{2{\mathrm{x}}^2-{\mathrm{x}}^2-1}{{\mathrm{x}}^2}$

$k^{\prime }(x)=\frac{x^2-1}{x^2}$

${\mathrm{k}}^{\prime }(\mathrm{x})=1-\frac{1}{{\mathrm{x}}^2}$

5. Differentiate m(x) = sin(x²)

Solution: Using the chain rule $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$ ,

Let u = x2, so m(x) = sin(u),

$\frac{\mathrm{d}}{\mathrm{du}}\sin (\mathrm{u})=\cos (\mathrm{u})$

$\frac{d}{dx}\left(x^2\right)=2x$

Therefore, m'(x) = 2x cos(x2)

6. Differentiate p(x) = e^3x

Solution: Using the exponential rule $\frac{d}{dx}\left[e^{u(x)}\right]=e^{u(x)}\cdot u^{\prime }(x)$ ,

Let u(x) = 3x,

Then u'(x) = 3

Therefore,

p'(x) = e^3x ·3

p'(x) = 3e^3x

7. Differentiate q(x) = ln(x² + 1)

Solution: Using the chain rule and the logarithm rule $\frac{d}{dx}[\ln (u(x))]=\frac{1}{u(x)}\cdot u^{\prime }(x)$,

Let u(x) = x² + 1,

Then u'(x) = 2x

Therefore, $q^{\prime }(x)=\frac{1}{x^2+1}\cdot 2x$

$q^{\prime }(x)=\frac{2x}{x^2+1}$

8. Differentiate x² + y² = 25

Solution: To find $\frac{\mathrm{dy}}{\mathrm{dx}}$ , differentiate both sides with respect to x,

$\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^2\right]+\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{y}}^2\right]=\frac{\mathrm{d}}{\mathrm{dx}}[25]$

$2x+2y\frac{dy}{dx}=0$

Solving for $\frac{dy}{dx}$ ,

$2y\frac{dy}{dx}=-2x$

$\frac{dy}{dx}=-\frac{x}{y}$

9. Differentiate r(x) = x³ for higher order up to 3

Solution: First Derivative:

r'(x) = 3x²

Second Derivative:

$r^{\prime \prime }(x)=\frac{d}{dx}\left[3x^2\right]=6x$

Third Derivative:

$r^{\prime \prime \prime }(x)=\frac{d}{dx}[6x]=6$

7.0Derivative Function Calculus Practice Question

1. Find the derivative of the function f(x) = 5x⁴.
2. Determine the derivative of g(x) = 3x³ – 4x² + 7x – 2.
3. Compute the derivative of the function h(x) = x² cos(x).
4. Find the derivative of k(x)= $\frac{x^2+1}{x}$
5. Determine the derivative of the function m(x) = sin(3x²).
6. Find $\frac{dy}{dx}$
7. if x² + y² = 16.
8. Calculate the derivative of p(x) = e^4x.
9. Find the derivative of q(x) = ln (x² + 3).
10. Compute the derivative of r(x) = tan(x).
11. Determine the second derivative of s(x) = x³ – 3x² + 4x – 5.

8.0Solved Questions on Derivative Function Calculus

Q. What are the basic rules of differentiation?

Ans: The basic rules of differentiation include:

• Constant Rule: $\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{c}]=0$
• Power Rule:  $\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^{\mathrm{n}}\right]={\mathrm{nx}}^{\mathrm{n}-1}$
• Constant Multiple Rule:  $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$
• Sum Rule: $\frac{d}{dx}[f(x)+g(x)]=f^{\prime }(x)+g^{\prime }(x)$ 
• Difference Rule: $\frac{d}{dx}[f(x)-g(x)]=f^{\prime }(x)-g^{\prime }(x)$
• Product Rule: $\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$
• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{g(x{)}^2}$
• Chain Rule: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Q. How do you find the derivative of a trigonometric function?

Ans: The derivatives of the basic trigonometric functions are:

• $\frac{d}{dx}[\sin (x)]=\cos (x)$
• $\frac{d}{dx}[\cos (x)]=-\sin (x)$
• $\frac{d}{dx}[\tan (x)]={\sec }^2(x)$
• $\frac{d}{dx}[\mathrm{cosec}(x)]=-\mathrm{cosec}(x)\cot (x)$
• $\frac{d}{dx}[\sec (x)]=\sec (x)\tan (x)$
• $\frac{d}{dx}[\cot (x)]=-{\mathrm{cosec}}^2(x)$

Q. What is implicit differentiation?

Ans: Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. For example, if you have an equation involving both x and y, such as x² + y² = 1, you differentiate both sides with respect to x and solve for $\frac{dy}{dx}.$

Q. How do you differentiate exponential and logarithmic functions?

Ans: For exponential functions:

• $\frac{d}{dx}\left[e^x\right]=e^x$
• $\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{a}}^{\mathrm{x}}\right]={\mathrm{a}}^{\mathrm{x}}\ln (\mathrm{a})$

For logarithmic functions:

• $\frac{\mathrm{d}}{\mathrm{dx}}[\ln (\mathrm{x})]=\frac{1}{\mathrm{x}}$
• $\frac{d}{dx}\left[{\log }_a(x)\right]=\frac{1}{x\ln (a)}$

Q. How do you handle derivatives of inverse trigonometric functions?

Ans: The derivatives of the basic inverse trigonometric functions are:

• $\frac{d}{dx}[\arcsin (x)]=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[\arccos (x)]=-\frac{1}{\sqrt{1-x^2}}$
• $\frac{\mathrm{d}}{\mathrm{dx}}[\arctan (\mathrm{x})]=\frac{1}{1+{\mathrm{x}}^2}$
• $\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{arccot}(\mathrm{x})]=-\frac{1}{1+{\mathrm{x}}^2}$
• $\frac{d}{dx}[\mathrm{arcsec}(x)]=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}[\mathrm{arccsc}(x)]=-\frac{1}{|x|\sqrt{x^2-1}}$