We know that differentiation is the inverse process of integration. If we take an example of sin x + 6, its differentiation is cos x. However, if we integrate cos x, we get sin x. Explain.

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Solution

Differentiation of sin x + 6 = cos x
But integration of cos x = sin x + C
We always add a constant term C while performing the indefinite integration. The value of this C can be found by applying suitable conditions given in the problem. Here the value of C is 6.

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if f(x)=x cos x, find f"(x), or d2ydx2

Physics Calculus Differentiation

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Using the Product Rule, we have f'(x)=xddx(cos x)+cos xddx(x)=-xsinx+cosx
To find f"(x) we differentiate f'(x):
f"(x)=ddx(-x sin x+cos x)=-xddx(sin x)+sin x ddx(-x)+ddx(cos x)
=-x cos x-sin x-sin x = -x cos x-2 sin x

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We know that differentiation is the inverse process of integration. If we take an example of sin x + 6, its differentiation is cos x. However, if we integrate cos x, we get sin x. Explain.

Differentiation of sin x + 6 = cos x But integration of cos x = sin x + C We always add a constant term C while performing the indefinite integration. The value of this C can be found by applying suitable conditions given in the problem. Here the value of C is 6.

Differentiation of sin x + 6 = cos x
But integration of cos x = sin x + C
We always add a constant term C while performing the indefinite integration. The value of this C can be found by applying suitable conditions given in the problem. Here the value of C is 6.