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Question: If \( y = \cos (5 - 3t) \) then find \( \dfrac{{dy}}{{dt}} \)...

If y=cos⁡(5−3t)y = \cos (5 - 3t) then find dydt\dfrac{{dy}}{{dt}}

Explanation

Solution

Hint : In this the function y is given we have to find the derivative of the function. The differentiation of a function is defined as the derivative or rate of change of a function. The function is said to be differentiable if the limit exists. Here in this question we have to find derivative w.r.t, t.
The derivative of some functions is given by ddx(cos⁡x)=−sin⁡x\dfrac{d}{{dx}}(\cos x) = - \sin x , ddx(a)=0\dfrac{d}{{dx}}(a) = 0 and ddx(bx)=b\dfrac{d}{{dx}}(bx) = b

Complete step-by-step answer :
Consider the given function y=cos⁡(5−3t)y = \cos (5 - 3t) . Now we have to differentiate this function w.r.t, t to get the derivative of the function
We have y=cos⁡(5−3t)y = \cos (5 - 3t)
On differentiating,
dydt=ddt(cos⁡(5−3t))\dfrac{{dy}}{{dt}} = \dfrac{d}{{dt}}(\cos (5 - 3t))
Since the given function is a composite function. now we will check whether the function is composite or not function
Let us consider g(t)=(5−3t)g(t) = (5 - 3t) and h(t)=cos⁡(t)h(t) = \cos (t)
g∘h=h(g(t))g \circ h = h(g(t))
⇒g∘h=h(5−3t)\Rightarrow g \circ h = h(5 - 3t)
⇒g∘h=cos⁡(5−3t)\Rightarrow g \circ h = \cos (5 - 3t)
⇒g∘h=y\Rightarrow g \circ h = y
Since the given function is a composite function of two functions then we can use the chain rule of derivative to the given function and hence we can find the derivative of the function.
Therefore, by applying the chain rule of derivative to the function so we have
⇒dydt=−sin⁡(5−3t)ddt(5−3t)\Rightarrow \dfrac{{dy}}{{dt}} = - \sin (5 - 3t)\dfrac{d}{{dt}}(5 - 3t)
Since the derivative of cos⁡x\cos x is −sin⁡x- \sin x
⇒dydt=−sin⁡(5−3t)[ddt(5)−ddt(3t)]\Rightarrow \dfrac{{dy}}{{dt}} = - \sin (5 - 3t)\left[ {\dfrac{d}{{dt}}(5) - \dfrac{d}{{dt}}(3t)} \right]
⇒dydt=−sin⁡(5−3t).(−3)\Rightarrow \dfrac{{dy}}{{dt}} = - \sin (5 - 3t).( - 3)
Simplifying we have
⇒dydt=3sin⁡(5−3t)\Rightarrow \dfrac{{dy}}{{dt}} = 3\sin (5 - 3t)
The derivative of constant function is zero, therefore ddt(5)=0\dfrac{d}{{dt}}(5) = 0 and the derivative of ddt(at)=a\dfrac{d}{{dt}}(at) = a so we have ddt(3t)=3\dfrac{d}{{dt}}(3t) = 3
Therefore, we have dydt=3sin⁡(5−3t)\dfrac{{dy}}{{dt}} = 3\sin (5 - 3t)
Hence, we obtained the derivative.
So, the correct answer is “ 3sin⁡(5−3t)3\sin (5 - 3t) ”.

Note : The differentiation is the rate of change of a function at a point. We must know about the chain rule of derivatives. The function can be written as a composite of two functions, if the function can be written as a composite of two functions then we can apply the chain rule of derivative. Hence by using the derivative formulas we can solve the function and hence obtain the solution