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Let P(n) represent the statement that 'f(n) is divisible by 19'. For the basis step, I prove that P(1) is true: f(1) = 3^3(1)-2+ 2³⁽¹⁾⁺¹ = 19. 19 is divisible by 19 so P(1) is true. I...

Stationary points of an equation are found where the gradient of the tangent at this point equals zero. A diagram can illustrate this. To find them differentiate the given equation (which gives the gradie...

(1) Find the gradient using differentiation (2) If the gradient at (x₁,y₁) is m,y - y₁ = m(x - x₁)
(1) We differentiate the given equation:dy/dx = 2x + 3 <...

To answer this we will use implicit differentiation with respect to x. So start by differentiating each term. On the left hand side 0 differentiates to 0. On the right hand side 5x² differentia...

Since f(x) is a product of the two functions e^x and sin(x^2), we can use the product rule which states that if f(x)=g(x)h(x), then f'(x)=g'(x)h(x)+g(x)h'(x). Let g(x)=e^x and h(x)=sin(x^2). Since the dif...