Properties of Derivatives: Key Insights and Examples

When diving into the world of calculus, understanding the properties of derivatives is pivotal. Let's explore these properties through examples to illustrate their significance and applications.

1. Linearity of the Derivative
The derivative of a sum of functions is the sum of their derivatives. Similarly, the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This property simplifies the differentiation process, especially when dealing with complex functions.

Example:
Consider the functions $f\left(x\right)=3x^2$f(x)=3x2 and $g\left(x\right)=5x$g(x)=5x. The derivative of $f\left(x\right)+g\left(x\right)$f(x)+g(x) is: $\frac{d}{dx}\left[f\right(x)+g(x\left)\right]=\frac{d}{dx}[3x^2+5x]=\frac{d}{dx}\left[3x^2\right]+\frac{d}{dx}\left[5x\right]=6x+5.$dxd​[f(x)+g(x)]=dxd​[3x2+5x]=dxd​[3x2]+dxd​[5x]=6x+5.

2. Product Rule
When differentiating the product of two functions, the product rule is used. It states that: $\frac{d}{dx}\left[u\right(x)\cdot v(x\left)\right]=u^{\prime }\left(x\right)\cdot v\left(x\right)+u\left(x\right)\cdot v^{\prime }\left(x\right).$dxd​[u(x)⋅v(x)]=u′(x)⋅v(x)+u(x)⋅v′(x).

Example:
For $u\left(x\right)=x^2$u(x)=x2 and $v\left(x\right)=\sin \left(x\right)$v(x)=sin(x), the derivative is: $\frac{d}{dx}[x^2\cdot \sin (x\left)\right]=\frac{d}{dx}\left[x^2\right]\cdot \sin \left(x\right)+x^2\cdot \frac{d}{dx}[\sin (x\left)\right]=2x\cdot \sin \left(x\right)+x^2\cdot \cos \left(x\right).$dxd​[x2⋅sin(x)]=dxd​[x2]⋅sin(x)+x2⋅dxd​[sin(x)]=2x⋅sin(x)+x2⋅cos(x).

3. Quotient Rule
The quotient rule is applied when differentiating a function divided by another function: $\frac{d}{dx}\left[\frac{u\left(x\right)}{v\left(x\right)}\right]=\frac{u^{\prime }\left(x\right)\cdot v\left(x\right)-u\left(x\right)\cdot v^{\prime }\left(x\right)}{\left[v\right(x){]}^2}.$dxd​[v(x)u(x)​]=[v(x)]2u′(x)⋅v(x)−u(x)⋅v′(x)​.

Example:
For $u\left(x\right)=x^2$u(x)=x2 and $v\left(x\right)=x+1$v(x)=x+1, the derivative is: $\frac{d}{dx}\left[\frac{x^2}{x+1}\right]=\frac{\left(2x\right)\cdot (x+1)-\left(x^2\right)\cdot \left(1\right)}{(x+1{)}^2}=\frac{2x^2+2x-x^2}{(x+1{)}^2}=\frac{x^2+2x}{(x+1{)}^2}.$dxd​[x+1x2​]=(x+1)2(2x)⋅(x+1)−(x2)⋅(1)​=(x+1)22x2+2x−x2​=(x+1)2x2+2x​.

4. Chain Rule
The chain rule is used to find the derivative of a composite function. It states that: $\frac{d}{dx}\left[f\right(g\left(x\right)\left)\right]=f^{\prime }\left(g\right(x\left)\right)\cdot g^{\prime }\left(x\right).$dxd​[f(g(x))]=f′(g(x))⋅g′(x).

Example:
If $f\left(x\right)=\ln \left(x\right)$f(x)=ln(x) and $g\left(x\right)=x^2$g(x)=x2, then: $\frac{d}{dx}[\ln (x^2\left)\right]=\frac{1}{x^2}\cdot \frac{d}{dx}\left[x^2\right]=\frac{1}{x^2}\cdot 2x=\frac{2x}{x^2}=\frac{2}{x}.$dxd​[ln(x2)]=x21​⋅dxd​[x2]=x21​⋅2x=x22x​=x2​.

5. Higher-Order Derivatives
Derivatives can be taken multiple times to find higher-order derivatives. For a function $f\left(x\right)$f(x), the second derivative is: $f^{\prime \prime }\left(x\right)=\frac{d}{dx}\left[f^{\prime }\right(x\left)\right].$f′′(x)=dxd​[f′(x)].

Example:
For $f\left(x\right)=x^3$f(x)=x3, the first derivative is $f^{\prime }\left(x\right)=3x^2$f′(x)=3x2. The second derivative is: $f^{\prime \prime }\left(x\right)=\frac{d}{dx}\left[3x^2\right]=6x.$f′′(x)=dxd​[3x2]=6x.

6. Derivatives of Common Functions
Knowing the derivatives of basic functions helps in differentiating more complex functions. Some common derivatives are:

• The derivative of $e^x$ex is $e^x$ex.
• The derivative of $\sin \left(x\right)$sin(x) is $\cos \left(x\right)$cos(x).
• The derivative of $\cos \left(x\right)$cos(x) is $-\sin \left(x\right)$−sin(x).

Example:
If $f\left(x\right)=e^x\cdot \sin \left(x\right)$f(x)=ex⋅sin(x), using the product rule: $\frac{d}{dx}[e^x\cdot \sin (x\left)\right]=e^x\cdot \cos \left(x\right)+\sin \left(x\right)\cdot e^x=e^x(\cos (x)+\sin (x\left)\right).$dxd​[ex⋅sin(x)]=ex⋅cos(x)+sin(x)⋅ex=ex(cos(x)+sin(x)).

In conclusion, mastering the properties of derivatives allows for more efficient differentiation and better understanding of function behavior. Whether you're dealing with sums, products, or compositions of functions, these properties are fundamental tools in calculus.