diasenamorn5l

2022-05-03

What is the derivative of $$\frac{5+7x+{x}^{2}}{\mathrm{tan}(\pi x)}$$?

Patricia Duffy

Beginner2022-05-04Added 16 answers

Differentiate using the Quotient Rule which states that $$\frac{d}{dx}[\frac{f(x)}{g(x)}]$$ is $$\frac{g(x)\frac{d}{dx}[f(x)]-f(x)\frac{d}{dx}[g(x)]}{g(x{)}^{2}}$$

where f(x)=5+7+x and $$g(x)=\mathrm{tan}(\pi x)$$

$$\frac{\mathrm{tan}(\pi x)\frac{d}{dx}[5+7+{x}^{2}]-(5+7+{x}^{2})\frac{d}{dx}[\mathrm{tan}(\pi x)]}{{\mathrm{tan}}^{2}(\pi x)}$$

Differentiate.

$$\frac{\mathrm{tan}(\pi x)(0+0+2x)-(5+7+{x}^{2})\frac{d}{dx}[\mathrm{tan}(\pi x)]}{{\mathrm{tan}}^{2}(\pi x)}$$

Combine terms.

$$\frac{\mathrm{tan}(\pi x)(2x)-(5+7+{x}^{2})\frac{d}{dx}[\mathrm{tan}(\pi x)]}{{\mathrm{tan}}^{2}(\pi x)}$$

Differentiate using the chain rule, which states that $$\frac{d}{dx}[f(g(x))]$$ is f'(g(x))g'(x) where $$f(x)=\mathrm{tan}(x)$$ and $$g(x)=\pi x$$.

$$\frac{\mathrm{tan}(\pi x)(2x)-(5+7+{x}^{2})({\mathrm{sec}}^{2}(\pi x)\frac{d}{dx}[\pi x])}{{\mathrm{tan}}^{2}(\pi x)}$$

Differentiate.

$$\frac{\mathrm{tan}(\pi x)(2x)-(5+7+{x}^{2})(\pi {\mathrm{sec}}^{2}(\pi x))}{{\mathrm{tan}}^{2}(\pi x)}$$

Simplify.

$$\frac{2x\mathrm{tan}(\pi x)-12\pi {\mathrm{sec}}^{2}(\pi x)-\pi {x}^{2}{\mathrm{sec}}^{2}(\pi x)}{{\mathrm{tan}}^{2}(\pi x)}$$

where f(x)=5+7+x and $$g(x)=\mathrm{tan}(\pi x)$$

$$\frac{\mathrm{tan}(\pi x)\frac{d}{dx}[5+7+{x}^{2}]-(5+7+{x}^{2})\frac{d}{dx}[\mathrm{tan}(\pi x)]}{{\mathrm{tan}}^{2}(\pi x)}$$

Differentiate.

$$\frac{\mathrm{tan}(\pi x)(0+0+2x)-(5+7+{x}^{2})\frac{d}{dx}[\mathrm{tan}(\pi x)]}{{\mathrm{tan}}^{2}(\pi x)}$$

Combine terms.

$$\frac{\mathrm{tan}(\pi x)(2x)-(5+7+{x}^{2})\frac{d}{dx}[\mathrm{tan}(\pi x)]}{{\mathrm{tan}}^{2}(\pi x)}$$

Differentiate using the chain rule, which states that $$\frac{d}{dx}[f(g(x))]$$ is f'(g(x))g'(x) where $$f(x)=\mathrm{tan}(x)$$ and $$g(x)=\pi x$$.

$$\frac{\mathrm{tan}(\pi x)(2x)-(5+7+{x}^{2})({\mathrm{sec}}^{2}(\pi x)\frac{d}{dx}[\pi x])}{{\mathrm{tan}}^{2}(\pi x)}$$

Differentiate.

$$\frac{\mathrm{tan}(\pi x)(2x)-(5+7+{x}^{2})(\pi {\mathrm{sec}}^{2}(\pi x))}{{\mathrm{tan}}^{2}(\pi x)}$$

Simplify.

$$\frac{2x\mathrm{tan}(\pi x)-12\pi {\mathrm{sec}}^{2}(\pi x)-\pi {x}^{2}{\mathrm{sec}}^{2}(\pi x)}{{\mathrm{tan}}^{2}(\pi x)}$$

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