Computing Derivatives Exercises

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• We can think of z as z = ln y where y = x⁵ + 4x².

Using the chain rule,

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• We can think of z as z = y⁷ where y = sin x + cos x.

Using the chain rule,

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• We can think of z as  where y = x⁹ – cos x.

Using the chain rule,

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• We can think of z as z = e^y where y = 12x + 4.Using the chain rule,

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• There are two ways to do this.Way 1: We can think of z as z = ln y where .Using the chain rule,

Way 2: First, rewrite the function:

Then taking the derivative only requires the rule for constant multiples, therefore

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• p = f(s) and s = g(z), therefore p = f(s) = f(g(z)). s is the inside function, so it will be the one that "cancels out":

The input to the inside function, z, will occur only in denominators:

and the outermost variable, p, will occur only in numerators:

The chain rule says

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• The middle function is r, since q = f(r) = f(g(t)), therefore

The innermost variable is t:

and the outermost variable is q:

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• Since x = f(y) and y = g(z), the overall function is

x = f(y) = f(g(z)).

The inner function is y, the innermost variable is z, and the outermost variable is x. The chain rule says

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•  If u = e^s and s = 5 – 3x²,

the chain rule says

The derivative of u with respect to s is e^s and the derivative of s with respect to x is -6x, therefore we find

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• If s = t⁴ and t = r⁴ + 1, the chain rule says

The derivative of s with respect to t is 4t³ and the derivative of t with respect to r is 4r³, so we find

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• If m = sin n and z = ln m, the chain rule says

The derivative of z with respect to m is  and the derivative of m with respect to n is cos n. Putting it all together gives us

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•  If z = (x⁴ + 4x) and y = z³ + z⁵, the chain rule says

The derivative of y with respect to z is 3z² + 5z⁴ and the derivative of z with respect to x is 4x³ + 4, so

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• If  and q = tan p, the chain rule says

The derivative of r with respect to q is  and the derivative of q with respect to p is 

sec² p, therefore

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• Let y = x² + 1 so that we can think of r as sin y + cos y.

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• Let y = r² + 3r so that we can think of p as p = e^y. Then

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• Let m = z⁴ + z³ so we can think of y as ln m. Then

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• Let m = x² + 4x – 2 so we can think of y as . Then

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• Let y = 12sin t and think of q as 14^y. Then