4. f(x)=8x^5 - 7e^x - 12x^(-3)+15\n5. f(x)=ln(5x^4 - 2)\n6. f(x)=(x^3 - 1)/(x^3 + 1)\n7. f(x)=e^(-2x)(6…

Explanation:

Step1: Differentiate (f(x)=x\ln x + x^{2})

Use product - rule ((uv)^\prime = u^\prime v+uv^\prime) for (x\ln x) where (u = x), (v=\ln x), (u^\prime=1), (v^\prime=\frac{1}{x}), and power - rule ((x^n)^\prime=nx^{n - 1}) for (x^{2}). (f^\prime(x)=(x\ln x)^\prime+(x^{2})^\prime=(1\times\ln x+x\times\frac{1}{x}) + 2x=\ln x + 1+2x)

Step2: Find (f^\prime(2))

Substitute (x = 2) into (f^\prime(x)). (f^\prime(2)=\ln 2+1 + 2\times2=\ln 2+5)

Step3: Find (f(2))

Substitute (x = 2) into (f(x)). (f(2)=2\ln 2+2^{2}=2\ln 2 + 4)

Step4: Find the equation of the tangent line

The equation of a tangent line is (y - y_0=m(x - x_0)), where ((x_0,y_0)=(2,f(2))) and (m = f^\prime(2)). (y-(2\ln 2 + 4)=(\ln 2+5)(x - 2)) (y-(2\ln 2 + 4)=(\ln 2+5)x-2\ln 2 - 10) (y=(\ln 2+5)x-6)

Answer:

a) (f^\prime(2)=\ln 2 + 5) b) (y=(\ln 2+5)x-6)