Basic Exercises on Differentiation Rules and Structural Recognition

1Problem 1

Differentiate $$(x^2+1)\sin x$$(x^2+1)\sin x.

1.1Sample Solution

By the product rule, the derivative is

1.2Explanation

Both factors contribute to the change of a product: $$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$$(fg)'=f'g+fg', not $$f^{\prime }{g}^{\prime }$$f'g'.

2Problem 2

Differentiate $$\text{[PARSE ERROR: Undefined("Command("dfrac")")]}{x}^2+1x-1$$\dfrac{x^2+1}{x-1}.

2.1Sample Solution

The domain is $$x\ne 1$$x\ne1. By the quotient rule,

2.2Explanation

The domain of the original function is $$\mathbb{R}\setminus \left\{1\right\}$$\mathbb{R}\setminus\{1\}. This restriction remains in force after deriving a formula for the derivative.

3Problem 3

Differentiate $$e^{x^2}\cos x$$e^{x^2}\cos x.

3.1Sample Solution

The product rule and chain rule give

3.2Explanation

The derivative of the inner function $$x^2$$x^2 must multiply the derivative of the outer exponential function.

4Problem 4

For $$x>0$$x>0, differentiate $$f\left(x\right)=x^x$$f(x)=x^x.

4.1Sample Solution

Set $$y=x^x$$y=x^x. Since $$x>0$$x>0, taking logarithms gives

Differentiating both sides with respect to $$x$$x yields

Therefore,

4.2Explanation

Both the base and exponent are variable, so neither the ordinary power rule nor the exponential rule applies directly. Logarithmic differentiation converts the exponent into a product before differentiation.

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