What is the derivative of 5 + 7 x + x 2 tan ⁡ ( π x ) ?

Question

What is the derivative of             5      +      7      x      +              x                  2                            tan      ⁡      (      π      x      )      ?

Patricia Duffy · Accepted Answer

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{g (x)^{2}}

where f(x)=5+7+x and

g (x) = \tan (π x)

\frac{\tan (π x) \frac{d}{d x} [5 + 7 + x^{2}] - (5 + 7 + x^{2}) \frac{d}{d x} [\tan (π x)]}{\tan^{2} (π x)}

Differentiate.

\frac{\tan (π x) (0 + 0 + 2 x) - (5 + 7 + x^{2}) \frac{d}{d x} [\tan (π x)]}{\tan^{2} (π x)}

Combine terms.

\frac{\tan (π x) (2 x) - (5 + 7 + x^{2}) \frac{d}{d x} [\tan (π x)]}{\tan^{2} (π x)}

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

is f'(g(x))g'(x) where

f (x) = \tan (x)

and

g (x) = π x

.

\frac{\tan (π x) (2 x) - (5 + 7 + x^{2}) (\sec^{2} (π x) \frac{d}{d x} [π x])}{\tan^{2} (π x)}

Differentiate.

\frac{\tan (π x) (2 x) - (5 + 7 + x^{2}) (π \sec^{2} (π x))}{\tan^{2} (π x)}

Simplify.

\frac{2 x \tan (π x) - 12 π \sec^{2} (π x) - π x^{2} \sec^{2} (π x)}{\tan^{2} (π x)}

What is the derivative of 5 + 7 x + x

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