# Find dy/dx when (A) #y= sqrtlnx# , (B) y= arctan 5x ?

(A)

# d/dx sqrtlnx = 1/(2xsqrt(lnx)) # (B)

# d/dx arctan 5x = 5/(25x^2+1)#

We seek the derivatives of:

We require some standard derivatives:

# {: (ul("Function"), ul("Derivative"), ul("Notes")),

(f(x), f'(x),), (af(x), af'(x), a " constant"), (x^n, nx^(n-1), n " constant (Power rule)"), (tan^(-1)x, 1/(1+x^2), ), (lnx, 1/x, ), (f(g(x)), f'(g(x)) \ g'(x),"(Chain rule)" ) :} #

Part (A):

Applying the chain rule, we have:

Part (B):

Again, applying the chain rule, we have:

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(A) ( \frac{dy}{dx} = \frac{1}{2x\sqrt{\ln{x}}} )

(B) ( \frac{dy}{dx} = \frac{5}{1+(5x)^2} )

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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