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Define the derivative of a function f(x) and use this to calculate the derivative of f(x)=x^n for positive integer n.

f'(x)=lim(h->0) of [f(x+h)-f(x)]/h. In the case where f(x)=x^n, we have that f(x+h)-f(x)=h*nx^(n-1) + (h^2)*p(x) for a polynomial p(x), shown by binomially expanding f(x+h). Then dividing through by h and...
TD
Answered by Tutor302361 D. • Maths tutor
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Given that f(x) = 1/x - sqrt(x) + 3. Find f'(1).

We are given that f(x) = 1/x - sqrt(x) + 3. Here, f(x) represents a function. A function is a relation or expression involving one or more variables. We are able to differentiate f(x) in order to find the gr...
Answered by • Maths tutor
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y = 4x^3 - 5/x^2 find dy/dx

First rewrite the equation into an easier form:y = y = 4x^3 - 5x^-2dy/dx means differentiate y with respect to xSo when differentiating, you look at the first term and write the co-efficient of the x term ( ...
DA
Answered by Dennis A. • Maths tutor
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Integrate the following with respect to x, f(x)=xsin(x)

f(x)= x sin(x) >>>>>>>>>>>>>>>>>> integral[ u dv/dx ] dx= uv - integral[v* du/dx] dx let x=u and sin(x)=dv/dx >>>>>>>>>>&gt...
JC
Answered by Jamie C. • Maths tutor
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The function f is defined for all real values of x as f(x) = c + 8x - x^2, where c is a constant. Given that the range of f is f(x) <= 19, find the value of c. Given instead that ff(2) = 8, find the possible values of c.

We know -x^2 has a maximum point of 0 at x=0, and -x^2 dominates the behaviour of f. We complete the square on this eqn, so we have f(x) = -(x - 4)^2 + 16 + c. We note that g(x) = -(x - 4)^2 has a maximum of...
SK
Answered by Sanmay K. • Maths tutor
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