Written as $\frac{d^2 y}{d x^2}, \frac{d^5 y}{d x^3}, \ldots \ldots \ldots \frac{d^n y}{d x^n}$ respectively.
- The function notation is also given as $f^{\prime}(x), f^{\prime \prime}(x), \ldots \ldots . f^n(x)$
Example 2:
Find the $2^{\text {nd }}$ derivatives of the following functions
(i) $f(x)=x^2-10 x+25$
(ii) $y=x^6+4 x^2-\frac{3}{x}$
Answer (1)
Find the first derivative of f ( x ) = x 2 − 10 x + 25 , which is f ′ ( x ) = 2 x − 10 .
Find the second derivative of f ( x ) , which is f ′′ ( x ) = 2 .
Find the first derivative of y = x 6 + 4 x 2 − x 3 , which is d x d y = 6 x 5 + 8 x + x 2 3 .
Find the second derivative of y , which is d x 2 d 2 y = 30 x 4 + 8 − x 3 6 .
Explanation
Problem Setup We are asked to find the second derivatives of two functions:
(i) f ( x ) = x 2 − 10 x + 25
(ii) y = x 6 + 4 x 2 − x 3
First Derivative of f(x) Let's find the first and second derivatives of f ( x ) = x 2 − 10 x + 25 .
The first derivative, f ′ ( x ) , is found by applying the power rule to each term:
f ′ ( x ) = 2 x − 10
Second Derivative of f(x) Now, let's find the second derivative, f ′′ ( x ) , by differentiating f ′ ( x ) :
f ′′ ( x ) = 2
Rewriting the function y Next, we'll find the first and second derivatives of y = x 6 + 4 x 2 − x 3 . It's helpful to rewrite the function as y = x 6 + 4 x 2 − 3 x − 1 before differentiating.
First Derivative of y The first derivative, d x d y , is found by applying the power rule to each term:
d x d y = 6 x 5 + 8 x + 3 x − 2 = 6 x 5 + 8 x + x 2 3
Second Derivative of y Now, let's find the second derivative, d x 2 d 2 y , by differentiating d x d y :
d x 2 d 2 y = 30 x 4 + 8 − 6 x − 3 = 30 x 4 + 8 − x 3 6
Final Answer Therefore, the second derivatives of the given functions are:
(i) f ′′ ( x ) = 2
(ii) d x 2 d 2 y = 30 x 4 + 8 − x 3 6
Examples
Understanding derivatives is crucial in physics, especially in kinematics. For instance, if f ( x ) represents the position of an object at time x , then f ′ ( x ) gives the object's velocity, and f ′′ ( x ) gives its acceleration. In this problem, finding the second derivative helps us understand how the rate of change of velocity (acceleration) behaves for different functions, which is essential for analyzing motion.
