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)
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Notes on Continuity and differentiability
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:
