Derivatives, marginal functions and elasticity

Compute from the definition the derivative of $f(x)=x^2+3x$ at the point $x_0=1$.

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Write the equation of the tangent line to the graph of $f(x)=e^x$ at the point with abscissa $0$.

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Compute the derivative of $f(x)=(2x-1)(x^2+3)$.

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Compute the derivative of $f(x)=\frac{x-1}{x+2}$.

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Compute the derivative of $f(x)=e^x\ln x$.

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Compute the derivative of $f(x)=(x^2-3x{)}^4$.

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Compute the derivative of $f(x)=\sqrt{x^2+9}$.

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The production cost is $C(q)=2000+15q+0.05q^2$. Compute the marginal cost for $q=50$ and the actual cost of the fifty-first unit.

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The inverse demand for a good is $p=60-0.3q$. Find the marginal revenue and the quantity at which it vanishes.

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The demand for a good is $q(p)=400-8p$. Compute the elasticity at the price $p=30$ and state whether a price increase raises revenue.

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Compute the elasticity of the demand $q(p)=500p^{-0.8}$, for $p>0$.

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Find the point $c$ guaranteed by Lagrange’s theorem for $f(x)=x^3$ on the interval $[1,3]$.

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Compute $\underset{x\to 0}{\lim }\frac{e^{3x}-1-3x}{x^2}$.

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Find the intervals of increase and decrease of $f(x)=x{e}^{-x}$.

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