Differentiate the following w.r.t. x :
(i) e−x
(ii) sin(logx),x>0
(iii) cos−1(ex)
(iv) ecosx
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Step-by-Step Solution
Step 1: Differentiate e−x
To differentiate e−x, we use the chain rule. The derivative of eu with respect to x is eu⋅dxdu. Here, u=−x. We differentiate e−x with respect to −x, and then multiply by the derivative of −x with respect to x.
Step 2: Simplify the derivative of e−x
The derivative of −x with respect to x is −1. Multiplying e−x by −1 gives us the final derivative.
Step 3: Differentiate sin(logx)
For sin(logx), we again use the chain rule. The derivative of sinu is cosu⋅dxdu. Here, u=logx. We differentiate sin(logx) with respect to logx, and then multiply by the derivative of logx with respect to x.
Step 4: Simplify the derivative of sin(logx)
The derivative of logx with respect to x is x1. Multiplying cos(logx) by x1 gives the final derivative.
Step 5: Differentiate cos−1(ex)
We use the chain rule for cos−1(ex). The derivative of cos−1u is 1−u2−1⋅dxdu. Here, u=ex. We differentiate cos−1(ex) with respect to ex, and then multiply by the derivative of ex with respect to x.
Step 6: Simplify the derivative of cos−1(ex)
The derivative of ex with respect to x is ex. Substituting this and simplifying gives the final derivative.
Step 7: Differentiate ecosx
For ecosx, we use the chain rule. The derivative of eu is eu⋅dxdu. Here, u=cosx. We differentiate ecosx with respect to cosx, and then multiply by the derivative of cosx with respect to x.
Step 8: Simplify the derivative of ecosx
The derivative of cosx with respect to x is −sinx. Multiplying ecosx by −sinx gives the final derivative.