What is the nth Derivative of cosx? [Solved]

The nth derivative of cosx is equal to cos(nπ/2 +x). The nth derivative of cos x is denoted by dⁿ/dxⁿ (cosx), and its formula is given as follows:

$\boxed{\dfrac{d^n}{dx^n}\left( \cos x\right)=\cos \left(\dfrac{n \pi}{2}+x \right)}$

nth Derivative of cos x

Question: Find the nth derivative of cosx.

Answer:

To find the nth derivative of cosx with respect to x, let us put

y = cosx.

Its first derivative is given by

y₁ = -sinx.

⇒ y₁ = cos$\left(\dfrac{\pi}{2}+x \right)$ using the trigonometric formula cos(π/2 +θ) = -sinθ.

Differentiating y₁ with respect to x, we obtain that

y2 = -sin$\left(\dfrac{\pi}{2}+x \right)$. ⇒ y2 = cos$\left(\dfrac{\pi}{2}+\dfrac{\pi}{2}+x \right)$ by the above rule: cos(π/2 +θ) = -sinθ. ⇒ y2 = cos$\left(\dfrac{2\pi}{2}+x \right)$.

In a similar way as above, it follows that

y₃ = -sin$\left(\dfrac{2\pi}{2}+x \right)$ = cos$\left(\dfrac{3\pi}{2}+x \right)$.

y₄ = -sin$\left(\dfrac{3\pi}{2}+x \right)$ = cos$\left(\dfrac{4\pi}{2}+x \right)$.

Conclusion: By observing the patterns, we see that the nth derivative of cosx is equal to cos(nπ/2 +x). That is, dⁿ/dxⁿ (cosx) = cos(nπ/2 +x).

Also Read:

nth derivative of 1/x	nth derivative of xn
nth derivative of 1/(ax+b)	nth derivative of sinx

Solved Problems

Question 1: Find the nth derivative of cos2x.

Answer:

As the nth derivative of cosx is equal to cos(nπ/2 +x), by the chain rule of differentiation the nth derivative of cos2x will be calculated as follows:

$\dfrac{d^n}{dx^n}\left( \cos 2x\right)= \cos \left(\dfrac{n \pi}{2}+2x \right) \times \left (\dfrac{d}{dx}(2x) \right)^n$ $=2^n \cos \left(\dfrac{n \pi}{2}+2x \right)$

Thus, the nth derivative of cos2x is equal to 2ⁿ cos(nπ/2 +2x).

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