Differentiate the functions with respect to x.
\(cos\ x^3.sin^2(x^5)\)
Differentiate the functions with respect to x.
\(cos\ x^3.sin^2(x^5)\)
Solution and Explanation
The given function is cos x3 . sin2(x5)
\(\frac {d}{dx}\)[cos x3. sin2(x5)] = sin2(x5) . \(\frac {d}{dx}\)(cos x3) + cos x3 . \(\frac {d}{dx}\)[sin2(x5)]
= sin2(x5) . (- sin x3) . \(\frac {d}{dx}\)(x3) + cos x3 . 2 sin (x5) . \(\frac {d}{dx}\)(sin x5)
= -sin x3 sin2(x5) . 3x2 + 2 sin x5 cos x3 . cos x5. \(\frac {d}{dx}\)(x5)
= -3x2 sin x3 . sin2(x5) + 2 sin x5 cos x5 cos x3.5x4
= 10x4 sin x5 cos x5 cos x3 - 3x2 sinx3 cosx3 sin2(x5)
Top Questions on Continuity and differentiability
- If $\tan^{-1} (x^2 - y^2) = a$, where $a$ is a constant, then $\frac{dy}{dx}$ is:
- CBSE CLASS XII - 2025
- Mathematics
- Continuity and differentiability
- If the function f(x) = $\begin{cases}\frac{k\cos x}{\pi - 2x} & ; x \neq \frac{\pi}{2} \\ 3 & ; x = \frac{\pi}{2} \end{cases}$ is continuous at x = $\frac{\pi}{2}$, then k is equal to
- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
- Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).
- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
Match List-I with List-II
List-I List-II (A) \( f(x) = |x| \) (I) Not differentiable at \( x = -2 \) only (B) \( f(x) = |x + 2| \) (II) Not differentiable at \( x = 0 \) only (C) \( f(x) = |x^2 - 4| \) (III) Not differentiable at \( x = 2 \) only (D) \( f(x) = |x - 2| \) (IV) Not differentiable at \( x = 2, -2 \) only
Choose the correct answer from the options given below:- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
Match List-I with List-II
List-I List-II (A) \( f(x) = |x| \) (I) Not differentiable at \( x = -2 \) only (B) \( f(x) = |x + 2| \) (II) Not differentiable at \( x = 0 \) only (C) \( f(x) = |x^2 - 4| \) (III) Not differentiable at \( x = 2 \) only (D) \( f(x) = |x - 2| \) (IV) Not differentiable at \( x = 2, -2 \) only
Choose the correct answer from the options given below:- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
Questions Asked in CBSE CLASS XII exam
- Your school is organizing a road safety awareness workshop for students of class IX – XII. As the head boy of your school, draft a notice informing the students about the workshop. Include other necessary details. You are Ashna/Ashish. Put your notice in a box.
- CBSE CLASS XII - 2025
- Letter Writing
If \(\begin{vmatrix} 2x & 3 \\ x & -8 \\ \end{vmatrix} = 0\), then the value of \(x\) is:
- ‘What I want should not be confused.’ What clarification does Pablo Neruda give to his readers? (Keeping Quiet)
- CBSE CLASS XII - 2025
- Literature
- A ladder 13 m long is leaning against the wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 12 m away from the wall?
- CBSE CLASS XII - 2025
- CBSE Compartment XII - 2025
- Application of derivatives
- If $P(A) = \frac{1}{7}$, $P(B) = \frac{5}{7}$ and $P(A \cap B) = \frac{4}{7}$, then $P(A | B)$ is:
- CBSE CLASS XII - 2025
- Probability
Concepts Used:
Derivatives
Derivatives are defined as a function's changing rate of change with relation to an independent variable. When there is a changing quantity and the rate of change is not constant, the derivative is utilised. The derivative is used to calculate the sensitivity of one variable (the dependent variable) to another one (independent variable). Derivatives relate to the instant rate of change of one quantity with relation to another. It is beneficial to explore the nature of a quantity on a moment-to-moment basis.
Few formulae for calculating derivatives of some basic functions are as follows:
