IB Mathematics Analysis and Approaches Question Type 2: Computing Harder Derivatives from First Principles Exercises

Using the first principles definition, compute the derivative of $f(x)=\cos x\sin x$.

Using the definition of the derivative, find $f^{\prime }(x)$ for $f(x)=x^2\ln x$ for $x>0$.

Using first principles, find $f^{\prime }(x)$ for $f(x)=x^2{e}^{-x}$.

Using the definition of the derivative, compute $f^{\prime }(x)$ for $f(x)=x^3\cos x$.

Using first principles, compute the derivative of $f(x)={\cos }^{2}x$.

Find $f^{\prime }(x)$ for $f(x)={\sin }^{2}x$ from first principles.

Find $f^{\prime }(x)$ for $f(x)=x{e}^x$ using the definition of the derivative.

Using first principles, find the derivative of $f(x)=\frac{\sin x}{x}$ for $x\ne 0$.

Using the definition of the derivative (first principles), find $f^{\prime }(x)$ for $f(x)=x\sin x$.

Using first principles, find the derivative of $f(x)=\ln (x)\sin x$ for $x>0$.

Using first principles, compute the derivative of $f(x)=e^x\sin x$.

Using first principles, find $f^{\prime }(x)$ for $f(x)=x^2\cos x$.